Chapter 11

Definite Integrals

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Section 11.1The Definite Integral as the Limit of a Sum[15/47%]
Problem 4541

Using definite integrals, find the limits of the following sums:

lim⁡n→∞(1n+2n+3n+…+2n−1n)⋅1n. \lim _{n \rightarrow \infty } \left(\frac{1}{n}+\frac{2}{n}+\frac{3}{n}+\ldots +\frac{2n-1}{n}\right) \cdot \frac{1}{n} .
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Problem 4542

Using definite integrals, find the limits of the following sums:

lim⁡n→∞(13n4+23n4+33n4+…+(4n−1)3n4). \lim _{n \rightarrow \infty } \left(\frac{1^{3}}{n^{4}}+\frac{2^{3}}{n^{4}}+\frac{3^{3}}{n^{4}}+\ldots +\frac{(4n-1)^{3}}{n^{4}}\right) .
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Problem 4543

Using definite integrals, find the limits of the following sums:

lim⁡n→∞πn(sin⁡πan+2πan+…+(n−1)πan). \lim _{n \rightarrow \infty } \frac{\pi }{n} \left(\sin \frac{\pi a}{n}+\frac{2\pi a}{n}+\ldots +\frac{(n-1)\pi a}{n}\right) .
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Problem 4544

Using definite integrals, find the limits of the following sums:

lim⁡n→∞(1n+1+1n+2+…+12n). \lim _{n \rightarrow \infty } \left(\frac{1}{n+1}+\frac{1}{n+2}+\ldots +\frac{1}{2n}\right) .
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Problem 4545

Using definite integrals, find the limits of the following sums:

lim⁡n→∞(nn2+12+nn2+22+…+nn2+(n−1)2). \lim _{n \rightarrow \infty } \left(\frac{n}{n^{2}+1^{2}}+\frac{n}{n^{2}+2^{2}}+\ldots +\frac{n}{n^{2}+(n-1)^{2}}\right) .
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Problem 4546

Using definite integrals, find the limits of the following sums:

lim⁡n→∞(24n2−12+24n2−22+…+24n2−n2). \lim _{n \rightarrow \infty } \left(\frac{2}{\sqrt{4n^{2}-1^{2}}}+\frac{2}{\sqrt{4n^{2}-2^{2}}}+\ldots +\frac{2}{\sqrt{4n^{2}-n^{2}}}\right) .
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Problem 4547

Find

lim⁡n→∞(n+1)(n+2)…2nnn. \lim _{n \rightarrow \infty } \frac{\sqrt[n]{(n+1)(n+2)\ldots 2n}}{n} .
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Problem 4548

Find the value of the integral ∫01ex dx\int_{0}^{1} e^{x}\, dx by regarding it as the limit of a sum.

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Problem 4549

Find, for b>a>0b > a > 0, the value of the integral ∫abxn dx\int_{a}^{b} x^{n}\, dx by regarding it as the limit of the sum ∑xνnΔxν\sum x_{\nu }^{n} \Delta x_{\nu }, where the values xνx_{\nu } form a geometric progression. (Fermat.)

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Problem 4550

Starting from the partial-fraction decomposition:

nxn−1xn−1=1x−x0+1x−x1+…+1x−xn−1,xν=e2πνin, \frac{n x^{n-1}}{x^{n}-1} = \frac{1}{x-x_{0}}+\frac{1}{x-x_{1}}+\ldots +\frac{1}{x-x_{n-1}}, \quad x_{\nu } = e^{\frac{2\pi \nu i}{n}},

obtain the equality:

∫02πdφx−eφi={0,for ∣x∣<1;2πx,for ∣x∣>1. \int _{0}^{2\pi } \frac{d\varphi }{x-e^{\varphi i}} = \begin{cases} 0, & \text{for } \left|x\right| < 1; \\ \dfrac {2\pi }{x}, & \text{for } \left|x\right| > 1 . \end{cases}
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Problem 4551

Starting from the formula

x2n−1=(x2−1)∏k=1n−1(x2−2xcos⁡kπn+1), x^{2n}-1 = (x^{2}-1) \prod _{k=1}^{n-1} \left(x^{2}-2x\cos \frac{k\pi }{n}+1\right),

obtain the value of the Poisson integral:

∫0πln⁡(1−2xcos⁡φ+x2) dφ={0,for ∣x∣<1;πln⁡(x2),for ∣x∣>1. \int _{0}^{\pi } \ln (1-2x\cos \varphi +x^{2})\, d\varphi = \begin{cases} 0, & \text{for } \left|x\right| < 1; \\ \pi \ln (x^{2}), & \text{for } \left|x\right| > 1 . \end{cases}
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Problem 4552

From the partial-fraction decomposition:

2nx2n−1x2n−1=1x2−1+∑k=1n−12x−2cos⁡kπnx2−2xcos⁡kπn+1 \frac{2n x^{2n-1}}{x^{2n}-1} = \frac{1}{x^{2}-1}+\sum _{k=1}^{n-1} \frac{2x-2\cos \frac{k\pi }{n}}{x^{2}-2x\cos \frac{k\pi }{n}+1}

obtain the equality:

∫0πx−cos⁡φx2−2xcos⁡φ+1 dφ={0,for ∣x∣<1;πx,for ∣x∣>1. \int _{0}^{\pi } \frac{x-\cos \varphi }{x^{2}-2x\cos \varphi +1}\, d\varphi = \begin{cases} 0, & \text{for } \left|x\right| < 1; \\ \dfrac {\pi }{x}, & \text{for } \left|x\right| > 1 . \end{cases}
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Problem 4553

The function [u]\left[u\right] equals the integer part of the variable uu, i.e. the integer mm such that m⩽u<m+1m \leqslant u < m+1. In that case

[2u]−2[u]={0,if u−m<12;1,if 12⩽u−m<1. \left[2u\right]-2\left[u\right] = \begin{cases} 0, & \text{if } u-m < \dfrac {1}{2}; \\ 1, & \text{if } \dfrac {1}{2} \leqslant u-m < 1 . \end{cases}

Prove that for the function f(x)=[2x]−2[1x]f(x) = \left[\dfrac {2}{x}\right]-2\left[\dfrac {1}{x}\right] on the interval (0,1)(0,1) the following inequality holds:

∑ν=1nOνΔxν<3max⁡Δxν, \sum _{\nu =1}^{n} O_{\nu }\Delta x_{\nu } < 3\sqrt{\max \Delta x_{\nu }},

and hence the function f(x)f(x) is Riemann integrable.

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Problem 4554

The function f(x)f(x) is defined by:

{f(x)=0,if x is irrational;f(x)=1q,if x=pq, \begin{cases} f(x) = 0, & \text{if } x \text{ is irrational}; \\ f(x) = \dfrac {1}{q}, & \text{if } x = \dfrac {p}{q}, \end{cases}

where pp and qq are relatively prime integers. Prove that on the interval (0,1)(0,1) the following inequality holds:

∑ν=1nOνΔxν<2max⁡Δxν. \sum _{\nu =1}^{n} O_{\nu }\Delta x_{\nu } < 2\sqrt{\max \Delta x_{\nu }} .

Thus f(x)f(x) is integrable on the interval (0,1)(0,1), although it has a discontinuity at every point with rational abscissa.

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Problem 4555

The function f(x)f(x) is defined by f(x)=0f(x) = 0 for xx irrational and f(x)=1f(x) = 1 for xx rational. Prove that it is not integrable on any interval.

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Section 11.2Mean Value Theorems. Improper Integrals[42/38%]
Problem 4556

Prove the following formula:

0<∫01x191+x63 dx<120. 0 < \int _{0}^{1} \frac{x^{19}}{\sqrt[3]{1+x^{6}}}\, dx < \frac{1}{20} .
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Problem 4557

Prove the following formula:

0<∫0200e−5x dx20+x<0,01. 0 < \int _{0}^{200} \frac{e^{-5x}\, dx}{20+x} < 0{,}01 .
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Problem 4558

Prove the following formula:

−π4<∫0∞cos⁡ax4+x2 dx<π4. -\frac{\pi }{4} < \int _{0}^{\infty } \frac{\cos ax}{4+x^{2}}\, dx < \frac{\pi }{4} .
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Problem 4559

Prove the following formula:

0<∫2∞e−x2 dx<14e4. 0 < \int _{2}^{\infty } e^{-x^{2}}\, dx < \frac{1}{4e^{4}} .
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Problem 4560

Prove the following formula:

0<∫1020x2 dxx4+x+1<120. 0 < \int _{10}^{20} \frac{x^{2}\, dx}{x^{4}+x+1} < \frac{1}{20} .
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Problem 4561

Prove the following formula:

119<∫1∞1+x201+x40 dx<119+139. \frac{1}{19} < \int _{1}^{\infty } \frac{1+x^{20}}{1+x^{40}}\, dx < \frac{1}{19}+\frac{1}{39} .
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Problem 4562

Prove the following formula:

1<∫011+x201+x40 dx<1+142. 1 < \int _{0}^{1} \frac{1+x^{20}}{1+x^{40}}\, dx < 1+\frac{1}{42} .
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Problem 4563

Prove the following formula:

2019<∫0∞1+x201+x40 dx<2019+120. \frac{20}{19} < \int _{0}^{\infty } \frac{1+x^{20}}{1+x^{40}}\, dx < \frac{20}{19}+\frac{1}{20} .
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Problem 4564

Prove the following formula:

0<∫1∞e−xn dx<1ne;n>1. 0 < \int _{1}^{\infty } e^{-x^{n}}\, dx < \frac{1}{ne}; \quad n > 1 .
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Problem 4565

Prove the following formula:

1−1n<∫01e−xn dx<1;n>1. 1-\frac{1}{n} < \int _{0}^{1} e^{-x^{n}}\, dx < 1; \quad n > 1 .
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Problem 4566

Prove the following formula:

1−1n<∫0∞e−xn dx<1+1ne. 1-\frac{1}{n} < \int _{0}^{\infty } e^{-x^{n}}\, dx < 1+\frac{1}{ne} .
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Problem 4567

Using the identity

1100+x=1100−x100(100+x), \frac{1}{100+x} = \frac{1}{100}-\frac{x}{100(100+x)},

prove the equality:

∫0∞e−x dx100+x=0,01−0,0001θ;0<θ<1. \int _{0}^{\infty } \frac{e^{-x}\, dx}{100+x} = 0{,}01-0{,}0001\theta ; \quad 0 < \theta < 1 .
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Problem 4568

Using the identity

x3x4+x+1=1x−x+1x(x4+x+1), \frac{x^{3}}{x^{4}+x+1} = \frac{1}{x}-\frac{x+1}{x(x^{4}+x+1)},

prove the equality:

∫100200x3 dxx4+x+1=ln⁡2−θ3⋅106;0<θ<1. \int _{100}^{200} \frac{x^{3}\, dx}{x^{4}+x+1} = \ln 2-\frac{\theta }{3 \cdot 10^{6}}; \quad 0 < \theta < 1 .
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Problem 4569

Prove the equalities:

∫100200sin⁡πxx dx=θ100π;∣θ∣<1. \int _{100}^{200} \frac{\sin \pi x}{x}\, dx = \frac{\theta }{100\pi }; \quad \left|\theta \right| < 1 .
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Problem 4570

Prove the equalities:

∫abcos⁡xx dx=2θa;∣θ∣<1;b>a>0. \int _{a}^{b} \frac{\cos x}{\sqrt{x}}\, dx = \frac{2\theta }{\sqrt{a}}; \quad \left|\theta \right| < 1; \quad b > a > 0 .
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Problem 4571

Prove the equalities:

∫100200sin⁡πx2 dx=θ200π;0<θ<1. \int _{100}^{200} \sin \pi x^{2}\, dx = \frac{\theta }{200\pi }; \quad 0 < \theta < 1 .
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Problem 4572

Prove the equalities:

∫100200sin⁡πxx dx=0,005π2θπ3⋅106;∣θ∣<1. \int _{100}^{200} \frac{\sin \pi x}{x}\, dx = \frac{0{,}005}{\pi }\frac{2\theta }{\pi ^{3} \cdot 10^{6}}; \quad \left|\theta \right| < 1 .
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Problem 4573

Investigate the convergence of the following integrals:

∫0∞x2 dxx4+x+1. \int _{0}^{\infty } \frac{x^{2}\, dx}{x^{4}+x+1} .
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Problem 4574

Investigate the convergence of the following integrals:

∫0∞x2 dxx3+x+1. \int _{0}^{\infty } \frac{x^{2}\, dx}{x^{3}+x+1} .
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Problem 4575

Investigate the convergence of the following integrals:

∫0∞sin⁡ax dx. \int _{0}^{\infty } \sin ax\, dx .
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Problem 4576

Investigate the convergence of the following integrals:

∫0∞sin⁡xn dx,n>0. \int _{0}^{\infty } \sin x^{n}\, dx, \quad n > 0 .
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Problem 4577

Investigate the convergence of the following integrals:

∫0∞xne−x dx. \int _{0}^{\infty } x^{n} e^{-x}\, dx .
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Problem 4578

Investigate the convergence of the following integrals:

∫0∞xσ dxx2+1. \int _{0}^{\infty } \frac{x^{\sigma }\, dx}{x^{2}+1} .
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Problem 4579

Investigate the convergence of the following integrals:

∫0∞xσx+2x+1 dx. \int _{0}^{\infty } x^{\sigma } \frac{x+2}{x+1}\, dx .
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Problem 4580

Investigate the convergence of the following integrals:

∫0∞xmsin⁡xn dx. \int _{0}^{\infty } x^{m} \sin x^{n}\, dx .
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Problem 4581

Investigate the convergence of the following integrals:

∫01xm(1−x2)n dx. \int _{0}^{1} x^{m}(1-x^{2})^{n}\, dx .
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Problem 4582

Investigate the convergence of the following integrals:

∫0+∞e−ax2xn dx. \int _{0}^{+\infty } e^{-ax^{2}} x^{n}\, dx .
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Problem 4583

Investigate the convergence of the following integrals:

∫02dxln⁡x. \int _{0}^{2} \frac{dx}{\ln x} .
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Problem 4584

Investigate the convergence of the following integrals:

∫0∞dx1+(ln⁡x)n. \int _{0}^{\infty } \frac{dx}{1+(\ln x)^{n}} .
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Problem 4585

Investigate the convergence of the following integrals:

∫0πln⁡sin⁡xx dx. \int _{0}^{\pi } \frac{\ln \sin x}{\sqrt{x}}\, dx .
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Problem 4586

Investigate the convergence of the following integrals:

∫0∞xxe−xn dx. \int _{0}^{\infty } x^{x} e^{-x^{n}}\, dx .
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Problem 4587

Investigate the convergence of the following integrals:

∫−∞+∞xne−(x2+1x2) dx. \int _{-\infty }^{+\infty } x^{n} e^{-(x^{2}+\frac{1}{x^{2}})}\, dx .
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Problem 4588

Investigate the convergence of the following integrals:

∫0∞sin⁡(x+1x)dxx. \int _{0}^{\infty } \sin \left(x+\frac{1}{x}\right) \frac{dx}{\sqrt{x}} .
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Problem 4589

Prove that for a⩾0a \geqslant 0 the integrals ∫0∞f(x)e−ax dx\int_{0}^{\infty } f(x) e^{-ax}\, dx and ∫0∞f(x)e−ax2 dx\int_{0}^{\infty } f(x) e^{-ax^{2}}\, dx converge uniformly, if the integral ∫0∞f(x) dx\int_{0}^{\infty } f(x)\, dx converges.

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Problem 4590

Prove that lim⁡a→0∫0∞f(x)φ(x,a) dx=∫0∞f(x) dx\lim_{a \rightarrow 0} \int_{0}^{\infty } f(x)\varphi (x,a)\, dx = \int_{0}^{\infty } f(x)\, dx, if the integral on the right-hand side makes sense, and φ(x,a)\varphi (x,a) satisfies the conditions:

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(a)

φ(x,a)→0\varphi (x,a) \rightarrow 0 as a→0a \rightarrow 0 uniformly with respect to xx on any finite interval 0<x<N0 < x < N;

(b)

For all aa sufficiently close to zero, φ(x,a)\varphi (x,a) is monotonic with respect to xx and does not exceed some constant in absolute value.

Problem 4591

The integral ∫0∞f(x) dx\int_{0}^{\infty } f(x)\, dx is absolutely convergent. Prove that ∫0∞f(x)sin⁡nx dx→0\int_{0}^{\infty } f(x)\sin nx\, dx \rightarrow 0 as n→∞n \rightarrow \infty.

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Problem 4592

Prove that

lim⁡n→∞∫01n2xn−1(1−x) dx≠∫01lim⁡n→∞n2xn−1(1−x) dx. \lim _{n \rightarrow \infty } \int _{0}^{1} n^{2} x^{n-1}(1-x)\, dx \neq \int _{0}^{1} \lim _{n \rightarrow \infty } n^{2} x^{n-1}(1-x)\, dx .
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Problem 4593

Prove that lim⁡y→y0∫abf(x,y) dx=∫ablim⁡y→y0f(x,y) dx\lim_{y \rightarrow y_{0}} \int_{a}^{b} f(x,y)\, dx = \int_{a}^{b} \lim_{y \rightarrow y_{0}} f(x,y)\, dx, if f(x,y)→f(x,y0)f(x,y) \rightarrow f(x,y_{0}) uniformly with respect to xx.

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Problem 4594

Prove that lim⁡y→y0∫0∞f(x,y) dx=∫0∞lim⁡y→y0f(x,y) dx\lim_{y \rightarrow y_{0}} \int_{0}^{\infty } f(x,y)\, dx = \int_{0}^{\infty } \lim_{y \rightarrow y_{0}} f(x,y)\, dx, if f(x,y)→f(x,y0)f(x,y) \rightarrow f(x,y_{0}) uniformly with respect to xx on any finite interval (0,n)(0,n), and moreover ∣f(x,y)∣<F(x)\left|f(x,y)\right| < F(x), with ∫0∞F(x) dx\int_{0}^{\infty } F(x)\, dx convergent.

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Problem 4595

Prove that if the integral ∫0∞f(x) dx\int_{0}^{\infty } f(x)\, dx is absolutely convergent, then

lim⁡n→∞∫0∞f(x)∣sin⁡nx∣ dx=2π∫0∞f(x) dx. \lim _{n \rightarrow \infty } \int _{0}^{\infty } f(x) \left|\sin nx\right|\, dx = \frac{2}{\pi } \int _{0}^{\infty } f(x)\, dx .
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Problem 4596

Prove that lim⁡σ→0σ∫0∞e−σtf(t) dt=lim⁡t→∞f(t)\lim_{\sigma \rightarrow 0} \sigma \int_{0}^{\infty } e^{-\sigma t} f(t)\, dt = \lim_{t \rightarrow \infty } f(t), assuming that the integral on the left-hand side and the limit on the right-hand side make sense.

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Problem 4597

In the correspondence between Euler and Lagrange, the integral

∫0∞e−ax−e−bxx dx, \int _{0}^{\infty } \frac{e^{-ax}-e^{-bx}}{x}\, dx,

equal to ln⁡ba\ln \dfrac {b}{a} for b>a>0b > a > 0, is transformed as follows:

∫0∞e−ax−e−bxx dx=∫0∞e−ax dxx−∫0∞e−bx dxx. \int _{0}^{\infty } \frac{e^{-ax}-e^{-bx}}{x}\, dx = \int _{0}^{\infty } \frac{e^{-ax}\, dx}{x}-\int _{0}^{\infty } \frac{e^{-bx}\, dx}{x} .

On the right-hand side, the first integral is transformed by the substitution ax=tax=t, and the second by the substitution bx=tbx=t. This gives:

ln⁡ba=∫0∞e−ax−e−bxx dx=∫0∞e−t dtt−∫0∞e−t dtt=0. \ln \frac{b}{a} = \int _{0}^{\infty } \frac{e^{-ax}-e^{-bx}}{x}\, dx = \int _{0}^{\infty } \frac{e^{-t}\, dt}{t}-\int _{0}^{\infty } \frac{e^{-t}\, dt}{t} = 0 .

What is the reason for the absurd result obtained?

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Section 11.3Computing Definite Integrals by Integration and Substitution[26/8%]
Problem 4598

Prove the equalities:

∫0∞dx1+x3=2π33. \int _{0}^{\infty } \frac{dx}{1+x^{3}} = \frac{2\pi }{3\sqrt{3}} .
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Problem 4599

Prove the equalities:

∫01x2+1x4+1 dx=π22. \int _{0}^{1} \frac{x^{2}+1}{x^{4}+1}\, dx = \frac{\pi }{2\sqrt{2}} .
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Problem 4600

Prove the equalities:

∫−11dx(a−x)1−x2=πa2−1;a>1. \int _{-1}^{1} \frac{dx}{(a-x)\sqrt{1-x^{2}}} = \frac{\pi }{\sqrt{a^{2}-1}}; \quad a > 1 .
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Problem 4601

Prove the equalities:

∫−11x4 dx(x2+1)1−x2=π2(2−1). \int _{-1}^{1} \frac{x^{4}\, dx}{(x^{2}+1)\sqrt{1-x^{2}}} = \frac{\pi }{2}(\sqrt{2}-1) .
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Problem 4602

Prove the equalities:

∫abx−ab−x x dx=π8(b−a)(a+3b). \int _{a}^{b} \sqrt{\frac{x-a}{b-x}}\, x\, dx = \frac{\pi }{8}(b-a)(a+3b) .
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Problem 4603

Prove the equalities:

∫0ax2a2−x2 dx=πa416. \int _{0}^{a} x^{2}\sqrt{a^{2}-x^{2}}\, dx = \frac{\pi a^{4}}{16} .
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Problem 4604

Prove the equalities:

∫02πdxa+bcos⁡x=2∫0πdxa+bcos⁡x=2πa2−b2;∣b∣<a. \int _{0}^{2\pi } \frac{dx}{a+b\cos x} = 2\int _{0}^{\pi } \frac{dx}{a+b\cos x} = \frac{2\pi }{\sqrt{a^{2}-b^{2}}}; \quad \left|b\right| < a .
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Problem 4605

Prove the equalities:

∫−11dxx2−2xcos⁡α+1=2π2∣sin⁡α∣. \int _{-1}^{1} \frac{dx}{x^{2}-2x\cos \alpha +1} = \frac{2\pi }{2\left|\sin \alpha \right|} .
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Problem 4606

Prove the equalities:

∫0πsin⁡x dx1+2acos⁡x+a2={2for ∣a∣⩽1,2∣a∣for ∣a∣>1. \int _{0}^{\pi } \frac{\sin x\, dx}{\sqrt{1+2a\cos x+a^{2}}} = \begin{cases} 2 & \text{for } \left|a\right| \leqslant 1, \\ \dfrac {2}{\left|a\right|} & \text{for } \left|a\right| > 1 . \end{cases}
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Problem 4607

Prove the equalities:

∫0πsin⁡2x dx1+2acos⁡x+a2={π2for ∣a∣<1,π2a2for ∣a∣>1. \int _{0}^{\pi } \frac{\sin ^{2} x\, dx}{1+2a\cos x+a^{2}} = \begin{cases} \dfrac {\pi }{2} & \text{for } \left|a\right| < 1, \\ \dfrac {\pi }{2a^{2}} & \text{for } \left|a\right| > 1 . \end{cases}
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Problem 4608

Prove the equalities:

∫−11dx(1−2ax+a2)(1−2bx+b2)=1abln⁡1+ab1−ab; \int _{-1}^{1} \frac{dx}{\sqrt{(1-2ax+a^{2})(1-2bx+b^{2})}} = \frac{1}{\sqrt{ab}}\ln \frac{1+\sqrt{ab}}{1-\sqrt{ab}}; ∣a∣<1,∣b∣<1,ab>0. \left|a\right| < 1, \quad \left|b\right| < 1, \quad ab > 0 .
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Problem 4609

Prove the equalities:

∫0∞e−axcos⁡bx dx=aa2+b2,a>0. \int _{0}^{\infty } e^{-ax}\cos bx\, dx = \frac{a}{a^{2}+b^{2}}, \quad a > 0 .
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Problem 4610

Prove the equalities:

∫0∞e−axsin⁡bx dx=ba2+b2,a>0. \int _{0}^{\infty } e^{-ax}\sin bx\, dx = \frac{b}{a^{2}+b^{2}}, \quad a > 0 .
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Problem 4611

Prove the equalities:

∫0π/2dxa2sin⁡2x+b2cos⁡2x=π2ab. \int _{0}^{\pi /2} \frac{dx}{a^{2}\sin ^{2} x+b^{2}\cos ^{2} x} = \frac{\pi }{2ab} .
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Problem 4612

Prove the equalities:

∫0π/2dx(a2sin⁡2x+b2cos⁡2x)2=π4a2+b2a3b3. \int _{0}^{\pi /2} \frac{dx}{(a^{2}\sin ^{2} x+b^{2}\cos ^{2} x)^{2}} = \frac{\pi }{4}\frac{a^{2}+b^{2}}{a^{3}b^{3}} .
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Problem 4613

Prove the equalities:

∫0π/2dxtg⁡x=∫0π/2tg⁡x dx=π2. \int _{0}^{\pi /2} \frac{dx}{\sqrt{\tg x}} = \int _{0}^{\pi /2} \sqrt{\tg x}\, dx = \frac{\pi }{\sqrt{2}} .
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Problem 4614

Using Euler's formulas

cos⁡x=exi+e−xi2,sin⁡x=exi−e−xi2i, \cos x = \frac{e^{xi}+e^{-xi}}{2}, \qquad \sin x = \frac{e^{xi}-e^{-xi}}{2i},

prove the equalities (mm and nn are positive integers):

∫0πsin⁡2nx dx=π22n(2n)!(n!)2. \int _{0}^{\pi } \sin ^{2n} x\, dx = \frac{\pi }{2^{2n}}\frac{(2n)!}{(n!)^{2}} .
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Problem 4615

Using Euler's formulas, prove the equalities (mm and nn are positive integers):

∫0πcos⁡(4n+1)xcos⁡x dx=π. \int _{0}^{\pi } \frac{\cos (4n+1)x}{\cos x}\, dx = \pi .
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Problem 4616

Using Euler's formulas, prove the equalities (mm and nn are positive integers):

∫0πsin⁡mxsin⁡x dx={0for m=2n;πfor m=2n+1. \int _{0}^{\pi } \frac{\sin mx}{\sin x}\, dx = \begin{cases} 0 & \text{for } m=2n; \\ \pi & \text{for } m=2n+1 . \end{cases}
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Problem 4617

Using Euler's formulas, prove the equalities (mm and nn are positive integers):

∫0πcos⁡nxcos⁡nx dx=π2n. \int _{0}^{\pi } \cos ^{n} x \cos nx\, dx = \frac{\pi }{2^{n}} .
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Problem 4618

Using Euler's formulas, prove the equalities (mm and nn are positive integers):

∫0πcos⁡nxsin⁡nx dx=0. \int _{0}^{\pi } \cos ^{n} x \sin nx\, dx = 0 .
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Problem 4619

Using Euler's formulas, prove the equalities (mm and nn are positive integers):

∫0πcos⁡mxcos⁡nx dx={0,if m−n is odd;π2mCmm−n2,if m−n is even; \int _{0}^{\pi } \cos ^{m} x \cos nx\, dx = \begin{cases} 0, & \text{if } m-n \text{ is odd}; \\ \dfrac {\pi }{2^{m}} C_{m}^{\frac{m-n}{2}}, & \text{if } m-n \text{ is even}; \end{cases} Cmk=m!k!(m−k)!;m⩾k. C_{m}^{k} = \frac{m!}{k!(m-k)!}; \quad m \geqslant k .
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Problem 4620

Using Euler's formulas, prove the equalities (mm and nn are positive integers):

∫0∞e−axcos⁡(2m−1)xcos⁡x dx=(−1)m+1a+2∑n=1m−1(−1)m+n−1aa2+4n2;a>0. \int _{0}^{\infty } e^{-ax}\frac{\cos (2m-1)x}{\cos x}\, dx = \frac{(-1)^{m+1}}{a}+2\sum _{n=1}^{m-1} \frac{(-1)^{m+n-1}a}{a^{2}+4n^{2}}; \quad a > 0 .
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Problem 4621

In the following problems one should apply the formula for repeated integration by parts, which can be written in the form:

∫uv(n) dx=uv(n−1)−u′v(n−2)+u′′v(n−3)+…+(−1)νu(ν)v(n−ν−1)+(−1)ν+1∫u(ν+1)v(n−ν−1) dx. \int u v^{(n)}\, dx = u v^{(n-1)}-u'v^{(n-2)}+u''v^{(n-3)}+\ldots +(-1)^{\nu } u^{(\nu )} v^{(n-\nu -1)}+(-1)^{\nu +1} \int u^{(\nu +1)} v^{(n-\nu -1)}\, dx .

Prove the equalities:

∫−11Pn(x)f(x) dx=0,∫0∞e−xLn(x)f(x) dx=0, \int _{-1}^{1} P_{n}(x) f(x)\, dx = 0, \qquad \int _{0}^{\infty } e^{-x} L_{n}(x) f(x)\, dx = 0, ∫−∞∞e−x2Hn(x)f(x) dx=0, \int _{-\infty }^{\infty } e^{-x^{2}} H_{n}(x) f(x)\, dx = 0,

where f(x)f(x) is an arbitrary polynomial of degree n−1n-1, and the Legendre polynomials Pn(x)P_{n}(x), the Laguerre polynomials Ln(x)L_{n}(x), and the Chebyshev--Hermite polynomials Hn(x)H_{n}(x) are defined by the equalities:

Pn(x)=12nn!dn(x2−1)ndxn;Ln(x)=exdnxne−xdxn; P_{n}(x) = \frac{1}{2^{n} n!} \frac{d^{n}(x^{2}-1)^{n}}{dx^{n}}; \quad L_{n}(x) = e^{x} \frac{d^{n} x^{n} e^{-x}}{dx^{n}}; Hn(x)=ex2dne−x2dxn. H_{n}(x) = e^{x^{2}} \frac{d^{n} e^{-x^{2}}}{dx^{n}} .
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Problem 4622

Find, for integer n>0n > 0, the value of the Euler integral

Γ(n)=∫0∞xn−1e−x dx. \Gamma (n) = \int _{0}^{\infty } x^{n-1} e^{-x}\, dx .
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Problem 4623

Find, for integers p>0p > 0 and q>0q > 0, the value of the Euler integral

B(p,q)=∫01xp−1(1−x)q−1 dx. B(p,q) = \int _{0}^{1} x^{p-1}(1-x)^{q-1}\, dx .
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Section 11.4Finding Integrals by Means of Reduction Formulas[20/90%]
Problem 4624

Find the values of the integrals (mm and nn are positive integers):

∫0π/4tg⁡2nx dx. \int _{0}^{\pi /4} \tg ^{2n} x\, dx .
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Problem 4625

Find the values of the integrals (mm and nn are positive integers):

∫01(1−x2)n dx. \int _{0}^{1} (1-x^{2})^{n}\, dx .
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Problem 4626

Find the values of the integrals (mm and nn are positive integers):

∫0∞dx(a2+x2)n. \int _{0}^{\infty } \frac{dx}{(a^{2}+x^{2})^{n}} .
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Problem 4627

Find the values of the integrals (mm and nn are positive integers):

∫−∞∞dx(Ax2+2Bx+C)n,AC−B2>0. \int _{-\infty }^{\infty } \frac{dx}{(Ax^{2}+2Bx+C)^{n}}, \quad AC-B^{2} > 0 .
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Problem 4628

Find the values of the integrals (mm and nn are positive integers):

∫0π/2sin⁡2nx dx. \int _{0}^{\pi /2} \sin ^{2n} x\, dx .
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Problem 4629

Find the values of the integrals (mm and nn are positive integers):

∫0π/2sin⁡2n+1x dx. \int _{0}^{\pi /2} \sin ^{2n+1} x\, dx .
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Problem 4630

Find the values of the integrals (mm and nn are positive integers):

∫0π/2sin⁡2n+1xcos⁡2m+1x dx. \int _{0}^{\pi /2} \sin ^{2n+1} x \cos ^{2m+1} x\, dx .
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Problem 4631

Using the known equality ∫0∞e−x2 dx=12π\int_{0}^{\infty } e^{-x^{2}}\, dx = \frac{1}{2}\sqrt{\pi }, find the value of the integral ∫0∞e−x2x2n dx\int_{0}^{\infty } e^{-x^{2}} x^{2n}\, dx.

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Problem 4632

For the Legendre, Laguerre, and Chebyshev--Hermite polynomials mentioned in problem 4621, prove the equalities:

∫−11Pn2(x) dx=22n+1,∫0∞e−xLn2(x) dx=(n!)2, \int _{-1}^{1} P_{n}^{2}(x)\, dx = \frac{2}{2n+1}, \qquad \int _{0}^{\infty } e^{-x} L_{n}^{2}(x)\, dx = (n!)^{2}, ∫−∞∞e−x2Hn2(x) dx=2nn!π. \int _{-\infty }^{\infty } e^{-x^{2}} H_{n}^{2}(x)\, dx = 2^{n} n! \sqrt{\pi } .
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Problem 4633

Find the integrals (mm and nn positive integers).

∫01xm(ln⁡x)n dx. \int _{0}^{1} x^{m}(\ln x)^{n}\, dx .
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Problem 4634

Find the integrals (mm and nn positive integers).

∫0∞xne−xsin⁡x dx. \int _{0}^{\infty } x^{n} e^{-x} \sin x\, dx .
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Problem 4635

Find the integrals (mm and nn positive integers).

∫0∞xne−xcos⁡x dx. \int _{0}^{\infty } x^{n} e^{-x} \cos x\, dx .
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Problem 4636

Remark. In the two preceding problems it is useful to add together two integrals whose mm differs by one or two units. In the two following problems it is useful to subtract two integrals whose mm differs by two units, and to apply integration by parts twice.

Find the integral:

∫0π/4cos⁡mxcos⁡x dx=um. \int _{0}^{\pi /4} \frac{\cos mx}{\cos x}\, dx = u_{m} .
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Problem 4637

Find the integral:

∫0π/2cos⁡(2m−1)xcos⁡x dx. \int _{0}^{\pi /2} \frac{\cos (2m-1)x}{\cos x}\, dx .
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Problem 4638

Find the integrals (mm and nn positive integers).

∫0∞e−axsin⁡2nx dx,a>0. \int _{0}^{\infty } e^{-ax} \sin ^{2n} x\, dx, \quad a > 0 .
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Problem 4639

Find the integrals (mm and nn positive integers).

∫0∞e−axsin⁡2n+1x dx,a>0. \int _{0}^{\infty } e^{-ax} \sin ^{2n+1} x\, dx, \quad a > 0 .
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Problem 4640

Find the integrals (mm and nn positive integers).

∫−π/2π/2e−mxcos⁡2n+1x dx. \int _{-\pi /2}^{\pi /2} e^{-mx} \cos ^{2n+1} x\, dx .
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Problem 4641

Find the integrals (mm and nn positive integers).

∫0∞e−ax2cos⁡(2n+1)xcos⁡x dx,a>0. \int _{0}^{\infty } e^{-ax^{2}} \frac{\cos (2n+1)x}{\cos x}\, dx, \quad a > 0 .
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Problem 4642

Prove the equality:

∫0π/2sin⁡mxsin⁡x dx=π2−2cos⁡mπ2∫01xm dx1+x2, \int _{0}^{\pi /2} \frac{\sin mx}{\sin x}\, dx = \frac{\pi }{2}-2\cos \frac{m\pi }{2} \int _{0}^{1} \frac{x^{m}\, dx}{1+x^{2}},

by verifying it for m=0m=0 and m=1m=1 and establishing that as mm increases by two units, both sides of the equality change by the same amount.

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Problem 4643

Find the integral Um=∫0π/2ln⁡cos⁡x⋅cos⁡2mx dxU_{m} = \int_{0}^{\pi /2} \ln \cos x \cdot \cos 2mx\, dx, by establishing a linear relation between UmU_{m} and Um−1U_{m-1}.

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Section 11.5Integration by Means of Series[44/75%]
Problem 4644

Compute the values of the integrals to within 10−510^{-5}:

∫01e−x2 dx. \int _{0}^{1} e^{-x^{2}}\, dx .
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Problem 4645

Compute the values of the integrals to within 10−510^{-5}:

∫01sin⁡xx dx. \int _{0}^{1} \frac{\sin x}{x}\, dx .
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Problem 4646

Compute the values of the integrals to within 10−510^{-5}:

∫5∞dxx4−1. \int _{5}^{\infty } \frac{dx}{\sqrt{x^{4}-1}} .
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Problem 4647

Compute the values of the integrals to within 10−510^{-5}:

∫100∞x dxx4−1. \int _{100}^{\infty } \frac{\sqrt{x}\, dx}{\sqrt{x^{4}-1}} .
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Problem 4648

Compute the values of the integrals to within 10−510^{-5}:

∫3∞x dxex−1. \int _{3}^{\infty } \frac{x\, dx}{e^{x}-1} .
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Problem 4649

Compute the values of the integrals to within 10−510^{-5}:

∫01/2ln⁡1+x21−x2dxx2. \int _{0}^{1/2} \ln \frac{1+x^{2}}{1-x^{2}} \frac{dx}{x^{2}} .
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Problem 4650

Prove the equality:

∫1/23/2ln⁡xx−1 dx=∫−1/21/2ln⁡(1+t)t dt. \int _{1/2}^{3/2} \frac{\ln x}{x-1}\, dx = \int _{-1/2}^{1/2} \frac{\ln (1+t)}{t}\, dt .
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Problem 4651

Express, as a series in powers of the eccentricity, the length of the contour of the ellipse x=acos⁡tx = a\cos t, y=bsin⁡ty = b\sin t.

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Problem 4652

The speed of a heavy point that has slid down some curve from height hh to height yy equals 2g(h−y)\sqrt{2g(h-y)}. Therefore the period of oscillation of a pendulum of length ll, swinging with maximum deflection angle α\alpha, is expressed by the formula

T=4l2g∫0αdφcos⁡φ−cos⁡α=4l2g∫0αdφ2(sin⁡2α2−sin⁡2φ2). T = 4\sqrt{\frac{l}{2g}} \int _{0}^{\alpha } \frac{d\varphi }{\sqrt{\cos \varphi -\cos \alpha }} = 4\sqrt{\frac{l}{2g}} \int _{0}^{\alpha } \frac{d\varphi }{\sqrt{2\left(\sin ^{2}\frac{\alpha }{2}-\sin ^{2}\frac{\varphi }{2}\right)}} .

From this, after the substitution sin⁡φ2=sin⁡α2sin⁡ψ\sin \dfrac {\varphi }{2} = \sin \dfrac {\alpha }{2}\sin \psi, one obtains:

T=4lg∫0π/2dψ1−σ2sin⁡2ψ;σ=sin⁡α2. T = 4\sqrt{\frac{l}{g}} \int _{0}^{\pi /2} \frac{d\psi }{\sqrt{1-\sigma ^{2}\sin ^{2}\psi }}; \quad \sigma = \sin \frac{\alpha }{2} .

Expanding in powers of σ\sigma and integrating term by term, obtain the formula:

T=2πlg(1+(12)2sin⁡2α2+(1⋅32⋅4)2sin⁡4α2+(1⋅3⋅52⋅4⋅6)2sin⁡6α2+…). T = 2\pi \sqrt{\frac{l}{g}} \left(1+\left(\frac{1}{2}\right)^{2}\sin ^{2}\frac{\alpha }{2}+\left(\frac{1 \cdot 3}{2 \cdot 4}\right)^{2}\sin ^{4}\frac{\alpha }{2}+\left(\frac{1 \cdot 3 \cdot 5}{2 \cdot 4 \cdot 6}\right)^{2}\sin ^{6}\frac{\alpha }{2}+\ldots \right) .
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Problem 4653

The following problems are based on knowing the three Euler series:

∑n=1∞1n2=π26;∑n=0∞1(2n+1)2=π28;∑n=1∞(−1)n−1n2=π212. \sum _{n=1}^{\infty } \frac{1}{n^{2}} = \frac{\pi ^{2}}{6}; \qquad \sum _{n=0}^{\infty } \frac{1}{(2n+1)^{2}} = \frac{\pi ^{2}}{8}; \qquad \sum _{n=1}^{\infty } \frac{(-1)^{n-1}}{n^{2}} = \frac{\pi ^{2}}{12} .

Find the value of the integral:

∫01ln⁡x dx1−x2. \int _{0}^{1} \frac{\ln x\, dx}{1-x^{2}} .
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Problem 4654

Find the value of the integral:

∫01ln⁡1+x1−xdxx. \int _{0}^{1} \ln \frac{1+x}{1-x} \frac{dx}{x} .
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Problem 4655

Find the integral:

∫01ln⁡(1+x)x dx. \int _{0}^{1} \frac{\ln (1+x)}{x}\, dx .
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Problem 4656

Find the integral:

∫01ln⁡(1−x)x dx. \int _{0}^{1} \frac{\ln (1-x)}{x}\, dx .
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Problem 4657

Find the integral:

∫0∞x dxex−1. \int _{0}^{\infty } \frac{x\, dx}{e^{x}-1} .
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Problem 4658

Find the integral:

∫0∞x dxex+1. \int _{0}^{\infty } \frac{x\, dx}{e^{x}+1} .
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Problem 4659

Find the integral:

∫0∞x dxeax−e−ax. \int _{0}^{\infty } \frac{x\, dx}{e^{ax}-e^{-ax}} .
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Problem 4660

Prove the equality:

∫0π/2(ln⁡tg⁡x)2 dx=2∫0π/4(ln⁡tg⁡x)2 dx. \int _{0}^{\pi /2} (\ln \tg x)^{2}\, dx = 2\int _{0}^{\pi /4} (\ln \tg x)^{2}\, dx .
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Problem 4661

Prove that

∫01arctg⁡xx dx=1−132+152−172+192−… \int _{0}^{1} \frac{\arctg x}{x}\, dx = 1-\frac{1}{3^{2}}+\frac{1}{5^{2}}-\frac{1}{7^{2}}+\frac{1}{9^{2}}-\ldots
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Problem 4662

Prove the equality, valid for any aa:

2∫0∞e−x2sin⁡ax dx=∑n=0∞(−1)nn!(2n+1)!a2n+1. 2\int _{0}^{\infty } e^{-x^{2}} \sin ax\, dx = \sum _{n=0}^{\infty } \frac{(-1)^{n} n!}{(2n+1)!} a^{2n+1} .
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Problem 4663

Using the formula ∫0∞e−x2x2n dx=1⋅3…(2n−1)2n+1π\int_{0}^{\infty } e^{-x^{2}} x^{2n}\, dx = \dfrac {1 \cdot 3 \ldots (2n-1)}{2^{n+1}}\sqrt{\pi }, prove the equality:

∫0∞e−x2cos⁡2ax dx=π2∑n=0∞(−1)nn!a2n=π2e−a2. \int _{0}^{\infty } e^{-x^{2}} \cos 2ax\, dx = \frac{\sqrt{\pi }}{2} \sum _{n=0}^{\infty } \frac{(-1)^{n}}{n!} a^{2n} = \frac{\sqrt{\pi }}{2} e^{-a^{2}} .

(Laplace.)

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Problem 4664

Using the identity

1a+x=1a−xa2+x2a3−…−x2n−1a2n+x2na2n(a+x), \frac{1}{a+x} = \frac{1}{a}-\frac{x}{a^{2}}+\frac{x^{2}}{a^{3}}-\ldots -\frac{x^{2n-1}}{a^{2n}}+\frac{x^{2n}}{a^{2n}(a+x)},

prove the formula

∫0∞e−x dxa+x=1a−1!a2+2!a3−…−(2n−1)!a2n+θ⋅(2n)!a2n+1;0<θ<1, \int _{0}^{\infty } \frac{e^{-x}\, dx}{a+x} = \frac{1}{a}-\frac{1!}{a^{2}}+\frac{2!}{a^{3}}-\ldots -\frac{(2n-1)!}{a^{2n}}+\frac{\theta \cdot (2n)!}{a^{2n+1}}; \quad 0 < \theta < 1,

which is very convenient for computing the integral on the left-hand side when aa is positive and large.

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Problem 4665

Taylor's formula with the Lagrange remainder for the function (1+x)−m(1+x)^{-m} is valid for any x>−1x > -1. Using this, prove the equality:

∫0∞e−x dxxm=e−a(1am−mam+1+m(m+1)am+2−…+(−1)nm(m+1)…(m+n−1)am+n+θR), \int _{0}^{\infty } \frac{e^{-x}\, dx}{x^{m}} = e^{-a}\left(\frac{1}{a^{m}}-\frac{m}{a^{m+1}}+\frac{m(m+1)}{a^{m+2}}-\ldots +(-1)^{n}\frac{m(m+1)\ldots (m+n-1)}{a^{m+n}}+\theta R\right),

where 0<θ<10 < \theta < 1, R=(−1)n+1m(m+1)…(m+n)am+n+1R = (-1)^{n+1}\dfrac {m(m+1)\ldots (m+n)}{a^{m+n+1}}; m>0m > 0, a>0a > 0.

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Problem 4666

Prove the equality:

∫0∞e−x2 dx=e−a22a(1−12a2+1⋅3(2a2)2−…+(−1)n−11⋅3⋅5…(2n−1)(2a2)n+θR), \int _{0}^{\infty } e^{-x^{2}}\, dx = \frac{e^{-a^{2}}}{2a}\left(1-\frac{1}{2a^{2}}+\frac{1 \cdot 3}{(2a^{2})^{2}}-\ldots +(-1)^{n-1}\frac{1 \cdot 3 \cdot 5 \ldots (2n-1)}{(2a^{2})^{n}}+\theta R\right), 0<θ<1,a>0,R=(−1)n1⋅3⋅5…(2n+1)(2a2)n+1. 0 < \theta < 1, \quad a > 0, \quad R = (-1)^{n}\frac{1 \cdot 3 \cdot 5 \ldots (2n+1)}{(2a^{2})^{n+1}} .
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Problem 4667

The following problems are based on the series expansions of three functions:

sin⁡x1−2acos⁡x+a2=sin⁡x+asin⁡2x+a2sin⁡3x+…, \frac{\sin x}{1-2a\cos x+a^{2}} = \sin x+a\sin 2x+a^{2}\sin 3x+\ldots , ln⁡(1−2acos⁡x+a2)=−2(acos⁡x+a22cos⁡2x+a33cos⁡3x+…), \ln (1-2a\cos x+a^{2}) = -2\left(a\cos x+\frac{a^{2}}{2}\cos 2x+\frac{a^{3}}{3}\cos 3x+\ldots \right), 1−a21−2acos⁡x+a2=1+2∑n=1∞ancos⁡nx. \frac{1-a^{2}}{1-2a\cos x+a^{2}} = 1+2\sum _{n=1}^{\infty } a^{n}\cos nx .

These expansions assume that ∣a∣<1\left|a\right| < 1.

Find the value of the following integral (mm and nn are positive integers):

∫0πcos⁡mx dx1−2acos⁡x+a2,a≠±1. \int _{0}^{\pi } \frac{\cos mx\, dx}{1-2a\cos x+a^{2}}, \quad a \neq \pm 1 .
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Problem 4668

Find the value of the following integral (mm and nn are positive integers):

∫0πsin⁡xsin⁡mx1−2acos⁡x+a2 dx. \int _{0}^{\pi } \frac{\sin x \sin mx}{1-2a\cos x+a^{2}}\, dx .
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Problem 4669

Find the value of the following integral (mm and nn are positive integers):

∫02πcos⁡mx1−acos⁡x dx,∣a∣<1.(Set a=2α1+α2.) \int _{0}^{2\pi } \frac{\cos mx}{1-a\cos x}\, dx, \quad \left|a\right| < 1 . \quad \text{(Set } a = \frac{2\alpha }{1+\alpha ^{2}} \text{.)}
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Problem 4670

Find the value of the following integral (mm and nn are positive integers):

∫02πcos⁡mx1−acos⁡x x dx,∣a∣<1. \int _{0}^{2\pi } \frac{\cos mx}{1-a\cos x}\, x\, dx, \quad \left|a\right| < 1 .
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Problem 4671

Find the value of the following integral (mm and nn are positive integers):

∫0πxsin⁡x1−2acos⁡x+a2 dx,∣a∣<1. \int _{0}^{\pi } \frac{x\sin x}{1-2a\cos x+a^{2}}\, dx, \quad \left|a\right| < 1 .
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Problem 4672

Find the value of the following integral (mm and nn are positive integers):

∫0πln⁡(1−2acos⁡x+a2) dx. \int _{0}^{\pi } \ln (1-2a\cos x+a^{2})\, dx .
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Problem 4673

Find the value of the following integral (mm and nn are positive integers):

∫02πsin⁡nxln⁡(1−2acos⁡x+a2) dx. \int _{0}^{2\pi } \sin nx \ln (1-2a\cos x+a^{2})\, dx .
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Problem 4674

Find the value of the following integral (mm and nn are positive integers):

∫0πln⁡(1−2acos⁡x+a2)cos⁡mx dx. \int _{0}^{\pi } \ln (1-2a\cos x+a^{2}) \cos mx\, dx .
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Problem 4675

Find the value of the following integral (mm and nn are positive integers):

∫0π/2ln⁡sin⁡x dx. \int _{0}^{\pi /2} \ln \sin x\, dx .
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Problem 4676

Find the value of the following integral (mm and nn are positive integers):

∫0πxln⁡sin⁡x dx. \int _{0}^{\pi } x\ln \sin x\, dx .
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Problem 4677

Find the value of the following integral (mm and nn are positive integers):

∫0π/2cos⁡2mxln⁡sin⁡x dx. \int _{0}^{\pi /2} \cos 2mx \ln \sin x\, dx .
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Problem 4678

Find the value of the following integral (mm and nn are positive integers):

∫0π(1−a2)(1−b2) dx(1−2acos⁡nx+a2)(1−2bcos⁡mx+b2); \int _{0}^{\pi } \frac{(1-a^{2})(1-b^{2})\, dx}{(1-2a\cos nx+a^{2})(1-2b\cos mx+b^{2})}; m and n are integers,∣a∣<1,∣b∣<1. m \text{ and } n \text{ are integers}, \quad \left|a\right| < 1, \quad \left|b\right| < 1 .
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Problem 4679

Find the value of the following integral (mm and nn are positive integers):

∫0πsin⁡2x dx(1−2acos⁡x+a2)(1−2bcos⁡x+b2);∣a∣<1,∣b∣<1. \int _{0}^{\pi } \frac{\sin ^{2} x\, dx}{(1-2a\cos x+a^{2})(1-2b\cos x+b^{2})}; \quad \left|a\right| < 1, \quad \left|b\right| < 1 .
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Problem 4680

Find the value of the following integral (mm and nn are positive integers):

∫0π/2ln⁡2sin⁡x dx. \int _{0}^{\pi /2} \ln ^{2}\sin x\, dx .
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Problem 4681

In the following three problems it is useful to apply two Laplace integrals:

∫0∞cos⁡ax dx1+x2=π2e−∣a∣;∫0∞xsin⁡ax dx1+x2=π2e−∣a∣sign⁡a. \int _{0}^{\infty } \frac{\cos ax\, dx}{1+x^{2}} = \frac{\pi }{2} e^{-\left|a\right|}; \qquad \int _{0}^{\infty } \frac{x\sin ax\, dx}{1+x^{2}} = \frac{\pi }{2} e^{-\left|a\right|}\operatorname {sign} a .

Find the integral:

∫0∞dx(1+x2)(1+2acos⁡mx+a2);∣a∣<1. \int _{0}^{\infty } \frac{dx}{(1+x^{2})(1+2a\cos mx+a^{2})}; \quad \left|a\right| < 1 .
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Problem 4682

Find the integral:

∫0∞xsin⁡mx(1+x2)(1+2acos⁡mx+a2) dx;∣a∣<1. \int _{0}^{\infty } \frac{x\sin mx}{(1+x^{2})(1+2a\cos mx+a^{2})}\, dx; \quad \left|a\right| < 1 .
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Problem 4683

Find the integral:

∫0∞ln⁡(1−2acos⁡x+a2)dx1+x2;∣a∣<1. \int _{0}^{\infty } \ln (1-2a\cos x+a^{2}) \frac{dx}{1+x^{2}}; \quad \left|a\right| < 1 .
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Problem 4684

In addition to power and trigonometric expansions, in some cases one uses expansions into series of rational fractions. Among formulas of this kind we note the following:

1sin⁡2x=∑n=−∞∞1(x+nπ)2;ctg⁡x=1x+∑n=1∞2xx2−π2n2; \frac{1}{\sin ^{2} x} = \sum _{n=-\infty }^{\infty } \frac{1}{(x+n\pi )^{2}}; \qquad \ctg x = \frac{1}{x}+\sum _{n=1}^{\infty } \frac{2x}{x^{2}-\pi ^{2} n^{2}}; 1sin⁡x=1x−2xx2−π2+2xx2−4π2−2xx2−9π2+…; \frac{1}{\sin x} = \frac{1}{x}-\frac{2x}{x^{2}-\pi ^{2}}+\frac{2x}{x^{2}-4\pi ^{2}}-\frac{2x}{x^{2}-9\pi ^{2}}+\ldots ; 1cos⁡x=−πx2−π24+3πx2−9π24−5πx2−25π24+…; \frac{1}{\cos x} = -\frac{\pi }{x^{2}-\frac{\pi ^{2}}{4}}+\frac{3\pi }{x^{2}-\frac{9\pi ^{2}}{4}}-\frac{5\pi }{x^{2}-\frac{25\pi ^{2}}{4}}+\ldots ; 1ex−1=1x−12+∑n=1∞2xx2+4π2n2; \frac{1}{e^{x}-1} = \frac{1}{x}-\frac{1}{2}+\sum _{n=1}^{\infty } \frac{2x}{x^{2}+4\pi ^{2} n^{2}}; 1sh⁡x=1x−2xx2+π2+2xx2+4π2−2xx2+9π2+…; \frac{1}{\operatorname {sh}x} = \frac{1}{x}-\frac{2x}{x^{2}+\pi ^{2}}+\frac{2x}{x^{2}+4\pi ^{2}}-\frac{2x}{x^{2}+9\pi ^{2}}+\ldots ; 1ex+1=12−∑n=1∞2xx2+(2n+1)2π2. \frac{1}{e^{x}+1} = \frac{1}{2}-\sum _{n=1}^{\infty } \frac{2x}{x^{2}+(2n+1)^{2}\pi ^{2}} .

With the help of the formulas indicated, the following equalities are proved:

2∫0∞sin⁡ux due2πu−1=1ex−1−1x+12. 2\int _{0}^{\infty } \frac{\sin ux\, du}{e^{2\pi u}-1} = \frac{1}{e^{x}-1}-\frac{1}{x}+\frac{1}{2} .
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Problem 4685

Using the formulas from problem 4684, prove the equality:

2∫0∞sin⁡ux dueu+1=1x−πsh⁡πx. 2\int _{0}^{\infty } \frac{\sin ux\, du}{e^{u}+1} = \frac{1}{x}-\frac{\pi }{\operatorname {sh}\pi x} .
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Problem 4686

Using the formulas from problem 4684, prove the equality:

2∫0∞tch⁡xteπt−1 dt=1sin⁡2x−1x2;∣x∣<π. 2\int _{0}^{\infty } \frac{t\operatorname {ch}xt}{e^{\pi t}-1}\, dt = \frac{1}{\sin ^{2}x}-\frac{1}{x^{2}}; \quad \left|x\right| < \pi .
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Problem 4687

Using the formulas from problem 4684, prove the equality:

∫0∞2sh⁡xteπt−1 dt=1x−ctg⁡x;∣x∣<π. \int _{0}^{\infty } \frac{2\operatorname {sh}xt}{e^{\pi t}-1}\, dt = \frac{1}{x}-\ctg x; \quad \left|x\right| < \pi .
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Section 11.6Differentiation and Integration Under the Integral Sign[32/88%]
Problem 4688

Starting from the equality ∫0∞e−ax dx=1a\int_{0}^{\infty } e^{-ax}\, dx = \dfrac {1}{a}, where a>0a > 0, find, for nn a positive integer, the value of the integral ∫0∞e−axxn−1 dx\int_{0}^{\infty } e^{-ax} x^{n-1}\, dx.

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Problem 4689

Starting from the equality ∫0∞dxx2+a2=π2a\int_{0}^{\infty } \dfrac {dx}{x^{2}+a^{2}} = \dfrac {\pi }{2a}, find the value of the integral ∫0∞dx(x2+a2)n\int_{0}^{\infty } \dfrac {dx}{(x^{2}+a^{2})^{n}}.

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Problem 4690

Starting from the equality ∫0∞e−ax2 dx=12πa\int_{0}^{\infty } e^{-ax^{2}}\, dx = \dfrac {1}{2}\sqrt{\dfrac {\pi }{a}}, a>0a > 0, find the integral

∫0∞e−ax2−e−bx2x2 dx \int _{0}^{\infty } \frac{e^{-ax^{2}}-e^{-bx^{2}}}{x^{2}}\, dx

for b>a>0b > a > 0.

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Problem 4691

Starting from the equality ∫0∞sin⁡axx dx=π2sign⁡a\int_{0}^{\infty } \dfrac {\sin ax}{x}\, dx = \dfrac {\pi }{2}\operatorname {sign} a, find the value of the integral ∫0∞cos⁡ax−cos⁡bxx2 dx\int_{0}^{\infty } \dfrac {\cos ax-\cos bx}{x^{2}}\, dx.

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Problem 4692

Find the values of the integrals:

∫0∞Ae−αx+Be−βx+Ce−γx+De−δxx dx;A+B+C+D=0,α>0,β>0,γ>0,δ>0. \int _{0}^{\infty } \frac{Ae^{-\alpha x}+Be^{-\beta x}+Ce^{-\gamma x}+De^{-\delta x}}{x}\, dx; \quad A+B+C+D=0, \quad \alpha > 0, \quad \beta > 0, \quad \gamma > 0, \quad \delta > 0 .
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Problem 4693

Find the values of the integrals:

∫0∞(e−αx−e−βxx)2 dx;α>0,β>0. \int _{0}^{\infty } \left(\frac{e^{-\alpha x}-e^{-\beta x}}{x}\right)^{2}\, dx; \quad \alpha > 0, \quad \beta > 0 .
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Problem 4694

Find the values of the integrals:

∫0∞e−αx−e−βxxsin⁡mx dx;α>0,β>0. \int _{0}^{\infty } \frac{e^{-\alpha x}-e^{-\beta x}}{x}\sin mx\, dx; \quad \alpha > 0, \quad \beta > 0 .
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Problem 4695

Find the values of the integrals:

∫0∞e−(x2+ax2) dx;a>0(Laplace.) \int _{0}^{\infty } e^{-\left(x^{2}+\frac{a}{x^{2}}\right)}\, dx; \quad a > 0 \quad \text{(Laplace.)}
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Problem 4696

Find the values of the integrals:

∫0∞(e−a2x2−e−b2x2) dx. \int _{0}^{\infty } \left(e^{-\frac{a^{2}}{x^{2}}}-e^{-\frac{b^{2}}{x^{2}}}\right)\, dx .
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Problem 4697

Find the values of the integrals:

∫0∞e−axsin⁡2bx dxx;a>0. \int _{0}^{\infty } e^{-ax} \sin ^{2} bx\, \frac{dx}{x}; \quad a > 0 .
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Problem 4698

Find the values of the integrals:

∫0∞e−axsin⁡2bx dxx2;a>0. \int _{0}^{\infty } e^{-ax} \sin ^{2} bx\, \frac{dx}{x^{2}}; \quad a > 0 .
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Problem 4699

Find the values of the integrals:

∫0∞xe−a2x2sin⁡2bx dx. \int _{0}^{\infty } x e^{-a^{2}x^{2}} \sin 2bx\, dx .
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Problem 4700

Find the values of the integrals:

∫0∞x2ne−x2cos⁡2βx dx,n a positive integer. \int _{0}^{\infty } x^{2n} e^{-x^{2}} \cos 2\beta x\, dx, \quad n \text{ a positive integer}.
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Problem 4701

Find the values of the integrals:

∫0∞e−ax2−cos⁡bxx2 dx;a>0,b>0. \int _{0}^{\infty } \frac{e^{-ax^{2}}-\cos bx}{x^{2}}\, dx; \quad a > 0, \quad b > 0 .
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Problem 4702

Find the values of the integrals:

∫0∞ln⁡(x2+a2)x2+b2 dx. \int _{0}^{\infty } \frac{\ln (x^{2}+a^{2})}{x^{2}+b^{2}}\, dx .
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Problem 4703

Find the values of the integrals:

∫01ln⁡(1−a2x2) dxx21−x2;a2<1. \int _{0}^{1} \frac{\ln (1-a^{2}x^{2})\, dx}{x^{2}\sqrt{1-x^{2}}}; \quad a^{2} < 1 .
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Problem 4704

Find the values of the integrals:

∫0αln⁡(1+αx) dx1+x2. \int _{0}^{\alpha } \frac{\ln (1+\alpha x)\, dx}{1+x^{2}} .
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Problem 4705

Find the values of the integrals:

∫01ln⁡(1−αx2)1−x2 dx;∣α∣<1. \int _{0}^{1} \frac{\ln (1-\alpha x^{2})}{\sqrt{1-x^{2}}}\, dx; \quad \left|\alpha \right| < 1 .
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Problem 4706

Find the values of the integrals:

∫01ln⁡(1−α2x2)x1−x2 dx;∣α∣<1. \int _{0}^{1} \frac{\ln (1-\alpha ^{2}x^{2})}{x\sqrt{1-x^{2}}}\, dx; \quad \left|\alpha \right| < 1 .
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Problem 4707

Find the values of the integrals:

∫0π/2ln⁡(cos⁡2x+m2sin⁡2x) dx. \int _{0}^{\pi /2} \ln (\cos ^{2} x+m^{2}\sin ^{2} x)\, dx .
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Problem 4708

Find the values of the integrals:

∫0π/2ln⁡1+asin⁡x1−asin⁡x dxsin⁡x;∣a∣<1. \int _{0}^{\pi /2} \ln \frac{1+a\sin x}{1-a\sin x}\, \frac{dx}{\sin x}; \quad \left|a\right| < 1 .
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Problem 4709

Find the values of the integrals:

∫0π/2ln⁡(1+acos⁡x)cos⁡x dx;∣a∣<1. \int _{0}^{\pi /2} \frac{\ln (1+a\cos x)}{\cos x}\, dx; \quad \left|a\right| < 1 .
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Problem 4710

Find the values of the integrals:

∫0π/2arctg⁡(atg⁡x)tg⁡x dx. \int _{0}^{\pi /2} \frac{\arctg (a\tg x)}{\tg x}\, dx .
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Problem 4711

Find the values of the integrals:

∫0∞arctg⁡αxx(1+β2x2) dx. \int _{0}^{\infty } \frac{\arctg \alpha x}{x(1+\beta ^{2}x^{2})}\, dx .
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Problem 4712

Find the values of the integrals:

∫0∞arctg⁡αx⋅arctg⁡βxx2 dx. \int _{0}^{\infty } \frac{\arctg \alpha x \cdot \arctg \beta x}{x^{2}}\, dx .
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Problem 4713

Find the values of the integrals:

∫0∞ln⁡(1+α2x2)⋅arctg⁡βxx3 dx. \int _{0}^{\infty } \frac{\ln (1+\alpha ^{2}x^{2}) \cdot \arctg \beta x}{x^{3}}\, dx .
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Problem 4714

Find the values of the integrals:

∫0∞ln⁡(1+a2x2)ln⁡(1+b2x2)x2 dx. \int _{0}^{\infty } \frac{\ln (1+a^{2}x^{2})\ln (1+b^{2}x^{2})}{x^{2}}\, dx .
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Problem 4715

Find the values of the integrals:

∫0∞cos⁡ax+cos⁡bx−2x2 dx. \int _{0}^{\infty } \frac{\cos ax+\cos bx-2}{x^{2}}\, dx .
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Problem 4716

Prove the equalities:

∫0xe−x2 dx=1π∫0∞e−t2sin⁡2txt dt. \int _{0}^{x} e^{-x^{2}}\, dx = \frac{1}{\sqrt{\pi }} \int _{0}^{\infty } e^{-t^{2}} \frac{\sin 2tx}{t}\, dt .
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Problem 4717

Prove the equalities:

∫0xet2 dt=ex2∫0∞e−t2sin⁡2xt dt. \int _{0}^{x} e^{t^{2}}\, dt = e^{x^{2}} \int _{0}^{\infty } e^{-t^{2}} \sin 2xt\, dt .
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Problem 4718

Prove the equalities:

∫0∞ln⁡(1+αx)1+x2 dx=π4ln⁡(1+α2)−∫0αln⁡α1+α2 dα;α>0. \int _{0}^{\infty } \frac{\ln (1+\alpha x)}{1+x^{2}}\, dx = \frac{\pi }{4}\ln (1+\alpha ^{2})-\int _{0}^{\alpha } \frac{\ln \alpha }{1+\alpha ^{2}}\, d\alpha ; \quad \alpha > 0 .
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Problem 4719

Prove the equalities:

(∫0xe−t2 dt)2=π4∫01e−x2(u2+1)1+u2 du. \left(\int _{0}^{x} e^{-t^{2}}\, dt\right)^{2} = \frac{\pi }{4} \int _{0}^{1} \frac{e^{-x^{2}(u^{2}+1)}}{1+u^{2}}\, du .
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Section 11.7Euler Integrals[41/76%]
Problem 4720

Another Euler integral, the beta function B(p,q)B(p,q), is expressed in terms of the gamma function. By definition, for p>0p > 0 and q>0q > 0 we have:

B(p,q)=∫01xp−1(1−x)q−1 dx. B(p,q) = \int _{0}^{1} x^{p-1}(1-x)^{q-1}\, dx .

There is a formula

B(p,q)=Γ(p)Γ(q)Γ(p+q). B(p,q) = \frac{\Gamma (p)\Gamma (q)}{\Gamma (p+q)} .

By a change of variables, integrals of the form

∫0∞xm(a+bxn)p dxfor a>0 and b>0 \int _{0}^{\infty } \frac{x^{m}}{(a+bx^{n})^{p}}\, dx \quad \text{for } a > 0 \text{ and } b > 0

reduce to the beta function. It is often useful to replace the fraction 1an\dfrac {1}{a^{n}}, where n>0n > 0, a>0a > 0, by means of the formula

1an=1Γ(n)∫0∞e−axxn−1 dx. \frac{1}{a^{n}} = \frac{1}{\Gamma (n)} \int _{0}^{\infty } e^{-ax} x^{n-1}\, dx .

Prove the equalities, for positive integers nn and mm:

∫0∞sin⁡2mxxp dx=(2m)!Γ(p)∫0∞tp−2 dt(t2+22)(t2+42)…(t2+4m2), \int _{0}^{\infty } \frac{\sin ^{2m} x}{x^{p}}\, dx = \frac{(2m)!}{\Gamma (p)} \int _{0}^{\infty } \frac{t^{p-2}\, dt}{(t^{2}+2^{2})(t^{2}+4^{2})\ldots (t^{2}+4m^{2})}, 1<p<2m+1. 1 < p < 2m+1 .
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Problem 4721

Prove the equalities, for positive integers nn and mm:

∫0∞sin⁡2m+1xxp dx= \int _{0}^{\infty } \frac{\sin ^{2m+1} x}{x^{p}}\, dx = =(2m+1)!Γ(p)∫0∞tp−1 dt(t2+1)(t2+32)…(t2+(2m+1)2), = \frac{(2m+1)!}{\Gamma (p)} \int _{0}^{\infty } \frac{t^{p-1}\, dt}{(t^{2}+1)(t^{2}+3^{2})\ldots (t^{2}+(2m+1)^{2})}, 1<p<2m+2. 1 < p < 2m+2 .
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Problem 4722

Prove the equalities, for positive integers nn and mm:

∫0∞sin⁡x dxx+a=∫0∞e−at dt1+t2,a>0. \int _{0}^{\infty } \frac{\sin x\, dx}{x+a} = \int _{0}^{\infty } \frac{e^{-at}\, dt}{1+t^{2}}, \quad a > 0 .
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Problem 4723

Prove the equalities, for positive integers nn and mm:

∫0∞cos⁡x dxx+a=∫0∞te−at dt1+t2,a>0. \int _{0}^{\infty } \frac{\cos x\, dx}{x+a} = \int _{0}^{\infty } \frac{t e^{-at}\, dt}{1+t^{2}}, \quad a > 0 .
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Problem 4724

Prove the equalities, for positive integers nn and mm:

∫0∞(1ex−1−1x+12)e−nxsin⁡nx dx= \int _{0}^{\infty } \left(\frac{1}{e^{x}-1}-\frac{1}{x}+\frac{1}{2}\right) e^{-nx} \sin nx\, dx = =4n2∫0∞x dx(x4+4n4)(e2πx−1). = 4n^{2} \int _{0}^{\infty } \frac{x\, dx}{(x^{4}+4n^{4})(e^{2\pi x}-1)} .
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Problem 4725

Find the value of the integrals:

∫0∞sin⁡2xx2 dx. \int _{0}^{\infty } \frac{\sin ^{2} x}{x^{2}}\, dx .
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Problem 4726

Find the value of the integrals:

∫0∞sin⁡4xx2 dx. \int _{0}^{\infty } \frac{\sin ^{4} x}{x^{2}}\, dx .
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Problem 4727

Find the value of the integrals:

∫0∞sin⁡3xx dx. \int _{0}^{\infty } \frac{\sin ^{3} x}{x}\, dx .
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Problem 4728

Find the value of the integrals:

∫0∞sin⁡3xx2 dx. \int _{0}^{\infty } \frac{\sin ^{3} x}{x^{2}}\, dx .
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Problem 4729

Find the value of the integrals:

∫0∞sin⁡4xx3 dx. \int _{0}^{\infty } \frac{\sin ^{4} x}{x^{3}}\, dx .
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Problem 4730

Find the value of the integrals:

∫01x1−x33 dx. \int _{0}^{1} x\sqrt[3]{1-x^{3}}\, dx .
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Problem 4731

Find the value of the integrals:

∫−11dx(1+x)2(1−x)3. \int _{-1}^{1} \frac{dx}{\sqrt[3]{(1+x)^{2}(1-x)}} .
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Problem 4732

Find the value of the integrals:

∫01x2n dxx(1−x)23. \int _{0}^{1} \frac{x^{2n}\, dx}{\sqrt[3]{x(1-x)^{2}}} .
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Problem 4733

Find the value of the integrals:

∫01dx1−xnn. \int _{0}^{1} \frac{dx}{\sqrt[n]{1-x^{n}}} .
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Problem 4734

Find the value of the integrals:

∫01dx1−x4. \int _{0}^{1} \frac{dx}{\sqrt{1-x^{4}}} .
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Problem 4735

Find the value of the integrals:

∫0∞x2α−11+x2 dx;0<α<1. \int _{0}^{\infty } \frac{x^{2\alpha -1}}{1+x^{2}}\, dx; \quad 0 < \alpha < 1 .
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Problem 4736

Find the value of the integrals:

∫0∞x dx(1+x3)2. \int _{0}^{\infty } \frac{x\, dx}{(1+x^{3})^{2}} .
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Problem 4737

Find the value of the integrals:

∫0∞x dx(1+x2)n. \int _{0}^{\infty } \frac{x\, dx}{(1+x^{2})^{n}} .
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Problem 4738

Find the value of the integrals:

∫0∞xm dx(a+bxn)p;a>0,b>0,np>m+1>0. \int _{0}^{\infty } \frac{x^{m}\, dx}{(a+bx^{n})^{p}}; \quad a > 0, \quad b > 0, \quad np > m+1 > 0 .
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Problem 4739

Find the value of the integrals:

∫0π/2tg⁡2a−1x dx;0<a<1. \int _{0}^{\pi /2} \tg ^{2a-1} x\, dx; \quad 0 < a < 1 .
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Problem 4740

Find the value of the integrals:

∫0π/2sin⁡nx dx;n>−1. \int _{0}^{\pi /2} \sin ^{n} x\, dx; \quad n > -1 .
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Problem 4741

Find the value of the integrals:

∫0π/2sin⁡m−1xcos⁡n−1x dx;m>0,n>0. \int _{0}^{\pi /2} \sin ^{m-1} x \cos ^{n-1} x\, dx; \quad m > 0, \quad n > 0 .
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Problem 4742

Find the value of the integrals:

∫0πsin⁡n−1φ dφ(1−ksin⁡φ)n;0<k<1,n>0. \int _{0}^{\pi } \frac{\sin ^{n-1}\varphi \, d\varphi }{(1-k\sin \varphi )^{n}}; \quad 0 < k < 1, \quad n > 0 .
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Problem 4743

Find the value of the integrals:

∫01xa−1−x−a1−x dx;0<a<1. \int _{0}^{1} \frac{x^{a-1}-x^{-a}}{1-x}\, dx; \quad 0 < a < 1 .
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Problem 4744

Find the value of the integrals:

∫01xα−1(1−x)β−1(x+a)α+β dx;α>0,β>0,a>0. \int _{0}^{1} \frac{x^{\alpha -1}(1-x)^{\beta -1}}{(x+a)^{\alpha +\beta }}\, dx; \quad \alpha > 0, \quad \beta > 0, \quad a > 0 .
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Problem 4745

Find the value of the integrals:

∫0∞xa−1−xb−1(1−x)ln⁡x dx. \int _{0}^{\infty } \frac{x^{a-1}-x^{b-1}}{(1-x)\ln x}\, dx .
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Problem 4746

Find the value of the integrals:

∫0∞xm−1 dx1+xn;m>0,n>m. \int _{0}^{\infty } \frac{x^{m-1}\, dx}{1+x^{n}}; \quad m > 0, \quad n > m .
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Problem 4747

Find the value of the integrals:

∫01xa−1(1+x)2 dx;0<a<2. \int _{0}^{1} \frac{x^{a-1}}{(1+x)^{2}}\, dx; \quad 0 < a < 2 .
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Problem 4748

Find the value of the integrals:

∫0∞xaln⁡x1+x2 dx;−1<a<1. \int _{0}^{\infty } \frac{x^{a}\ln x}{1+x^{2}}\, dx; \quad -1 < a < 1 .
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Problem 4749

Find the value of the integrals:

∫0∞xaln⁡2x1+x2 dx;a2<1. \int _{0}^{\infty } \frac{x^{a}\ln ^{2}x}{1+x^{2}}\, dx; \quad a^{2} < 1 .
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Problem 4750

Find the value of the integrals:

∫0∞sh⁡μxsh⁡νx dx;ν>μ>0. \int _{0}^{\infty } \frac{\operatorname {sh}\mu x}{\operatorname {sh}\nu x}\, dx; \quad \nu > \mu > 0 .
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Problem 4751

Find the value of the integrals:

∫0∞ch⁡μxch⁡νx dx;ν>μ>0. \int _{0}^{\infty } \frac{\operatorname {ch}\mu x}{\operatorname {ch}\nu x}\, dx; \quad \nu > \mu > 0 .
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Problem 4752

Find the value of the integrals:

∫01ln⁡Γ(x) dx. \int _{0}^{1} \ln \Gamma (x)\, dx .
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Problem 4753

Find the value of the integrals:

∫0a+1ln⁡Γ(x) dx;a>0. \int _{0}^{a+1} \ln \Gamma (x)\, dx; \quad a > 0 .
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Problem 4754

Prove the equalities:

∫0∞e−x4 dx∫0∞x2e−x4 dx=π82. \int _{0}^{\infty } e^{-x^{4}}\, dx \int _{0}^{\infty } x^{2} e^{-x^{4}}\, dx = \frac{\pi }{8\sqrt{2}} .
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Problem 4755

Prove the equalities:

∫01dx1−x4∫01x2 dx1−x4=π4. \int _{0}^{1} \frac{dx}{\sqrt{1-x^{4}}} \int _{0}^{1} \frac{x^{2}\, dx}{\sqrt{1-x^{4}}} = \frac{\pi }{4} .
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Problem 4756

Prove the equalities:

∫0∞xp−1cos⁡ax dx=1apΓ(p)cos⁡πp2;0<p<1. \int _{0}^{\infty } x^{p-1} \cos ax\, dx = \frac{1}{a^{p}}\Gamma (p)\cos \frac{\pi p}{2}; \quad 0 < p < 1 .
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Problem 4757

Prove the equalities:

∫0∞xp−1sin⁡ax dx=1apΓ(p)sin⁡πp2;−1<p<1. \int _{0}^{\infty } x^{p-1} \sin ax\, dx = \frac{1}{a^{p}}\Gamma (p)\sin \frac{\pi p}{2}; \quad -1 < p < 1 .
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Problem 4758

Prove the equalities:

∫0∞xp−12 dx(x2+ax+b)p=π2b(a+2b)p−32⋅Γ(p−12)Γ(p);b>0, \int _{0}^{\infty } \frac{x^{p-\frac{1}{2}}\, dx}{(x^{2}+ax+b)^{p}} = \frac{\sqrt{\pi }}{2\sqrt{b}\left(a+2\sqrt{b}\right)^{p-\frac{3}{2}}} \cdot \frac{\Gamma \left(p-\frac{1}{2}\right)}{\Gamma (p)}; \quad b > 0, a+2b>0,p>12. a+2\sqrt{b} > 0, \quad p > \frac{1}{2} .
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Problem 4759

Find the area bounded by the curve xn+yn=anx^{n}+y^{n} = a^{n}, for x>0x > 0, y>0y > 0, n>0n > 0.

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Problem 4760

Find the volume bounded by the surface xn+yn+zn=anx^{n}+y^{n}+z^{n} = a^{n}, for x>0x > 0, y>0y > 0, z>0z > 0, n>0n > 0.

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Section 11.8Miscellaneous Problems[96/53%]
Problem 4761

If F(x)=arctg⁡1xF(x) = \arctg \dfrac {1}{x}, then F′(x)=−11+x2F'(x) = -\dfrac {1}{1+x^{2}}. Is the equality ∫ab(−dx1+x2)=arctg⁡1b−arctg⁡1a\displaystyle \int_{a}^{b} \left(-\frac{dx}{1+x^{2}}\right) = \arctg \frac{1}{b}-\arctg \frac{1}{a} valid?

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Problem 4762

If F(x)=11+e1/xF(x) = \dfrac {1}{1+e^{1/x}}, then F′(x)=e1/xx2(1+e1/x)2=f(x)F'(x) = \dfrac {e^{1/x}}{x^{2}\left(1+e^{1/x}\right)^{2}} = f(x). When is the equality

∫abf(x) dx=F(b)−F(a)−1  ? \int _{a}^{b} f(x)\, dx = F(b)-F(a)-1 \; ?

valid?

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Problem 4763

Prove that ∫0∞(sin⁡2x)x3xα dx\int_{0}^{\infty } (\sin^{2}x)^{x^{3}} x^{\alpha }\, dx converges for 0⩽α⩽120 \leqslant \alpha \leqslant \dfrac {1}{2}, although the integrand is positive and does not tend to zero as x→∞x \rightarrow \infty, since for arbitrarily large values of xx it can take arbitrarily large values.

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Problem 4764

Prove the equalities:

lim⁡n→∞∫0∞e−xn dx=1. \lim _{n \rightarrow \infty } \int _{0}^{\infty } e^{-x^{n}}\, dx = 1 .
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Problem 4765

Prove the equalities:

lim⁡n→∞∫0∞1+xn1+x2n dx=1. \lim _{n \rightarrow \infty } \int _{0}^{\infty } \frac{1+x^{n}}{1+x^{2n}}\, dx = 1 .
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Problem 4766

Prove the equalities:

lim⁡n→∞n(1−∫0∞e−xn dx)=−Γ′(1). \lim _{n \rightarrow \infty } n\left(1-\int _{0}^{\infty } e^{-x^{n}}\, dx\right) = -\Gamma '(1) .
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Problem 4767

Prove the equalities:

lim⁡α→0α∫−∞∞f(x+u) duα2+u2=πf(x+0)+f(x−0)2, \lim _{\alpha \rightarrow 0} \alpha \int _{-\infty }^{\infty } \frac{f(x+u)\, du}{\alpha ^{2}+u^{2}} = \pi \frac{f(x+0)+f(x-0)}{2},

if the integral is absolutely convergent, and f(x+0)f(x+0) and f(x−0)f(x-0) are the right and left limits of f(ξ)f(\xi ) as ξ→x\xi \rightarrow x.

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Problem 4768

Prove the equality (Frullani's formula):

∫0∞f(ax)−f(bx)x dx=f(0)ln⁡ba,a>0,b>0, \int _{0}^{\infty } \frac{f(ax)-f(bx)}{x}\, dx = f(0)\ln \frac{b}{a}, \quad a > 0, \quad b > 0,

if for A>0A > 0 the integral ∫A∞f(x)x dx\int_{A}^{\infty } \dfrac {f(x)}{x}\, dx makes sense.

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Problem 4769

Prove that lim⁡a→∞an∫0∞f(x)cos⁡ax dx=0\lim_{a \rightarrow \infty } a^{n} \int_{0}^{\infty } f(x) \cos ax\, dx = 0, if f(x)f(x) is an even function for which the integral of the modulus of f(m)(x)f^{(m)}(x) over the interval (−∞,∞)(-\infty ,\infty ) makes sense for every m⩽n+1m \leqslant n+1.

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Problem 4770

Prove that lim⁡a→∞an∫0∞f(x)sin⁡ax dx=0\lim_{a \rightarrow \infty } a^{n} \int_{0}^{\infty } f(x) \sin ax\, dx = 0, if f(x)f(x) is an odd function, and the integral of the absolute value of f(m)(x)f^{(m)}(x), taken from −∞-\infty to +∞+\infty, makes sense for every m⩽n+1m \leqslant n+1.

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Problem 4771

Prove that f(x)=0f(x) = 0 for a<x<ba < x < b, if f(x)f(x) is continuous and for every integer n⩾0n \geqslant 0 the integral ∫abf(x)xn dx=0\int_{a}^{b} f(x) x^{n}\, dx = 0.

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Problem 4772

Prove that f(x)=0f(x) = 0 for 0⩽x⩽2π0 \leqslant x \leqslant 2\pi, if it is continuous, and for every positive integer nn (and for n=0n=0) the integrals ∫02πf(x)cos⁡nx dx\int_{0}^{2\pi } f(x)\cos nx\, dx and ∫02πf(x)sin⁡nx dx\int_{0}^{2\pi } f(x)\sin nx\, dx are equal to zero.

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Problem 4773

Prove that for f(x)=e−x4sin⁡xx4f(x) = e^{-\sqrt[4]{x}} \sin x \sqrt[4]{x}, for all integers n⩾0n \geqslant 0, the equality ∫0∞f(x)xn dx=0\int_{0}^{\infty } f(x) x^{n}\, dx = 0 holds.

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Problem 4774

Prove that for the function f(x)=1−cos⁡xxf(x) = \dfrac {1-\cos x}{x}, for any integer n≠1n \neq 1, the equalities hold:

∫−∞∞f(x)cos⁡nx dx=0,∫−∞∞f(x)sin⁡nx dx=0. \int _{-\infty }^{\infty } f(x) \cos nx\, dx = 0, \qquad \int _{-\infty }^{\infty } f(x) \sin nx\, dx = 0 .
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Problem 4775

Prove the equality:

∫01dxxα(x+h)1−α=ln⁡1h+A+O(h), \int _{0}^{1} \frac{dx}{x^{\alpha }(x+h)^{1-\alpha }} = \ln \frac{1}{h}+A+O(h),

where O(h)O(h) denotes a quantity whose ratio to hh remains bounded in absolute value as h→+0h \rightarrow +0 and 0<α<10 < \alpha < 1.

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Problem 4776

Prove that in absolute value the integral ∫abφ(x)eiμx dx\int_{a}^{b} \varphi (x) e^{i\mu x}\, dx does not exceed 2φ(a)μ\dfrac {2\varphi (a)}{\mu }, if a<ba < b, and φ(x)\varphi (x) is a decreasing positive function; μ>0\mu > 0.

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Problem 4777

Prove the inequality: ∫0∞f(x)sin⁡ax dx>0\int_{0}^{\infty } f(x) \sin ax\, dx > 0, if f(x)f(x) is a decreasing function; a>0a > 0 and the integral converges.

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Problem 4778

Prove the inequality: ∫0∞f(x)sin⁡ax dx>0\int_{0}^{\infty } f(x) \sin ax\, dx > 0, if f(x)f(x) is a decreasing function and the integral converges.

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Problem 4779

Prove that as h→0h \rightarrow 0 the equality holds

lim⁡h→01hπ∫−∞∞f(t)e−(t−x)2h2 dt=f(x), \lim _{h \rightarrow 0} \frac{1}{h\sqrt{\pi }} \int _{-\infty }^{\infty } f(t) e^{\frac{-(t-x)^{2}}{h^{2}}}\, dt = f(x),

if f(x)f(x) is a continuous function for which ∫−∞∞∣f(t)∣e−t2h2 dt\int_{-\infty }^{\infty } \left|f(t)\right| e^{-\frac{t^{2}}{h^{2}}}\, dt makes sense for sufficiently small ∣h∣\left|h\right|.

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Problem 4780

Prove Laplace's theorem: if f(x)f(x) has a finite derivative on the interval (a,b)(a,b), and φ(x)>0\varphi (x) > 0 attains its largest value at x=cx=c, where a<c<ba < c < b, then as n→∞n \rightarrow \infty the equality holds

∫abf(x)(φ(x))n dx=f(c)(φ(c))n−2πφ(c)nφ′′(c)(1+O(1n)), \int _{a}^{b} f(x) \left(\varphi (x)\right)^{n}\, dx = f(c)\left(\varphi (c)\right)^{n} \sqrt{-\frac{2\pi \varphi (c)}{n\varphi ''(c)}} \left(1+O\left(\frac{1}{\sqrt{n}}\right)\right),

where O(1n)O\left(\dfrac {1}{\sqrt{n}}\right) is, in absolute value, no greater than 1n\dfrac {1}{\sqrt{n}} multiplied by some constant. If cc coincides with one of the numbers aa or bb, then an extra factor of 12\dfrac {1}{2} must be placed on the right-hand side.

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Problem 4781

Prove the equalities:

∫01(1−x2)n dx=12πn+O(1n). \int _{0}^{1} (1-x^{2})^{n}\, dx = \frac{1}{2}\sqrt{\frac{\pi }{n}}+O\left(\frac{1}{n}\right) .
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Problem 4782

Prove the equalities:

∫0∞e−xxn−1 dx=nne−n2πn(1+O(1n)). \int _{0}^{\infty } e^{-x} x^{n-1}\, dx = n^{n} e^{-n} \sqrt{\frac{2\pi }{n}} \left(1+O\left(\frac{1}{\sqrt{n}}\right)\right) .
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Problem 4783

Prove the equalities:

∫0π/2cos⁡2nx1+x dx=12πn+O(1n). \int _{0}^{\pi /2} \frac{\cos ^{2n} x}{1+x}\, dx = \frac{1}{2}\sqrt{\frac{\pi }{n}}+O\left(\frac{1}{n}\right) .
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Problem 4784

Prove the equalities:

∫0π/2sin⁡2nx1+x dx=1π+2πn+O(1n). \int _{0}^{\pi /2} \frac{\sin ^{2n} x}{1+x}\, dx = \frac{1}{\pi +2}\sqrt{\frac{\pi }{n}}+O\left(\frac{1}{n}\right) .
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Problem 4785

Prove the inequality:

(∫abφ(x)ψ(x) dx)2⩽∫abφ2(x) dx∫abψ2(x) dx. \left(\int _{a}^{b} \varphi (x)\psi (x)\, dx\right)^{2} \leqslant \int _{a}^{b} \varphi ^{2}(x)\, dx \int _{a}^{b} \psi ^{2}(x)\, dx .

(Bunyakovsky.)

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Problem 4786

Prove the inequality:

φ(∫abp(x)f(x) dx∫abp(x) dx)⩽∫abp(x)φ(f(x)) dx∫abp(x) dx, \varphi \left(\frac{\int _{a}^{b} p(x) f(x)\, dx}{\int _{a}^{b} p(x)\, dx}\right) \leqslant \frac{\int _{a}^{b} p(x) \varphi \left(f(x)\right)\, dx}{\int _{a}^{b} p(x)\, dx},

where p(x)⩾0p(x) \geqslant 0, and the curve y=φ(x)y = \varphi (x) is convex downward.

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Problem 4787

Obtain the analogous inequality for the case when the curve y=φ(x)y = \varphi (x) is convex upward.

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Problem 4788

Prove that the function of tt equal to

(1b−a∫ab(f(x))t dx)1t,a<b, \left(\frac{1}{b-a} \int _{a}^{b} \left(f(x)\right)^{t}\, dx\right)^{\frac{1}{t}}, \quad a < b,

is nondecreasing as tt increases, if f(x)f(x) has a positive lower bound.

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Problem 4789

The moments of the function f(x)f(x) on the interval (a,b)(a,b) are the integrals

Mn=∫abf(x)xn dx. M_{n} = \int _{a}^{b} f(x) x^{n}\, dx .

Prove that if 0<a<b0 < a < b and f(x)>0f(x) > 0 for x>0x > 0, then the series ∑tnMn\sum \dfrac {t^{n}}{M_{n}} has a finite or infinite radius of convergence, according as the interval (a,b)(a,b) is finite or infinite.

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Problem 4790

By splitting the interval (−∞,∞)(-\infty ,\infty ) into the parts (−∞,x−h)(-\infty ,x-h), (x−h,x)(x-h,x), (x,x+h)(x,x+h), (x+h,∞)(x+h,\infty ), where hh is an arbitrarily chosen number, prove the equality:

lim⁡n→∞1π∫−∞∞f(u)sin⁡n(u−x)u−x du=f(x). \lim _{n \rightarrow \infty } \frac{1}{\pi } \int _{-\infty }^{\infty } f(u) \frac{\sin n(u-x)}{u-x}\, du = f(x) .

Here, for simplicity, it is assumed that f(x)f(x) is continuous everywhere, f′(x)f'(x) exists everywhere and is bounded on every given interval, and ∫−∞∞f(u) du\int_{-\infty }^{\infty } f(u)\, du is absolutely convergent.

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Problem 4791

Under the same conditions on f(x)f(x) as in the preceding problem, prove Fourier's formula:

f(x)=12π∫−∞∞du∫−∞∞f(v)cos⁡u(v−x) dv. f(x) = \frac{1}{2\pi } \int _{-\infty }^{\infty } du \int _{-\infty }^{\infty } f(v) \cos u(v-x)\, dv .
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Problem 4792

Show that the Fourier integral (see the preceding problem) remains valid also in the case when f(x)f(x) has finite discontinuities at isolated points, provided the earlier conditions are preserved at the remaining points. In that case, at the points of discontinuity the value of the integral equals 12(f(x+0)+f(x−0))\dfrac {1}{2}\left(f(x+0)+f(x-0)\right), where f(x+0)f(x+0) and f(x−0)f(x-0) are the limits from the right and from the left of f(ξ)f(\xi ) as ξ→x\xi \rightarrow x.

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Problem 4793

Prove that for functions φ(x)\varphi (x) satisfying the conditions of the preceding problem (that is, φ(x)\varphi (x) is continuous everywhere, φ′(x)\varphi '(x) exists everywhere and is bounded on every given interval, and ∫−∞∞φ(u) du\int_{-\infty }^{\infty } \varphi (u)\, du is absolutely convergent), the following theorem holds: if ψ(x)=12π∫−∞∞φ(t)eixt dt\psi (x) = \dfrac {1}{\sqrt{2\pi }} \int_{-\infty }^{\infty } \varphi (t) e^{ixt}\, dt, then φ(x)=12π∫−∞∞ψ(t)e−ixt dt\varphi (x) = \dfrac {1}{\sqrt{2\pi }} \int_{-\infty }^{\infty } \psi (t) e^{-ixt}\, dt.

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Problem 4794

Prove, for the functions φ(x)\varphi (x) and ψ(x)\psi (x) introduced in the preceding problem, the equality:

∫−∞∞∣φ(x)∣2 dx=∫−∞∞∣ψ(x)∣2 dx. \int _{-\infty }^{\infty } \left|\varphi (x)\right|^{2}\, dx = \int _{-\infty }^{\infty } \left|\psi (x)\right|^{2}\, dx .
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Problem 4795

The notation φ(x)∼a0xσ+a1xσ+1+a2xσ+2+…\varphi (x) \sim \dfrac {a_{0}}{x^{\sigma }}+\dfrac {a_{1}}{x^{\sigma +1}}+\dfrac {a_{2}}{x^{\sigma +2}}+\ldots is read: the function φ(x)\varphi (x) has the asymptotic series on the right-hand side; this notation means that as x→∞x \rightarrow \infty, for any ν\nu,

xσ+ν(φ(x)−a0xσ−a1xσ+1−…−aνxσ+ν)→0. x^{\sigma +\nu } \left(\varphi (x)-\frac{a_{0}}{x^{\sigma }}-\frac{a_{1}}{x^{\sigma +1}}-\ldots -\frac{a_{\nu }}{x^{\sigma +\nu }}\right) \rightarrow 0 .

Prove that

∫0∞e−xtf(t) dt∼a0x+a1⋅1!x2+a2⋅2!x3+a3⋅3!x4+…, \int _{0}^{\infty } e^{-xt} f(t)\, dt \sim \frac{a_{0}}{x}+\frac{a_{1} \cdot 1!}{x^{2}}+\frac{a_{2} \cdot 2!}{x^{3}}+\frac{a_{3} \cdot 3!}{x^{4}}+\ldots ,

if f(t)=a0+a1t+a2t2+…f(t) = a_{0}+a_{1}t+a_{2}t^{2}+\ldots and the integral converges.

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Problem 4796

Prove that as n→∞n \rightarrow \infty

∫01xn−1dx1−x∼πn+a1Γ(32)nn+a2Γ(52)n2n+…, \int _{0}^{1} x^{n-1} \frac{dx}{\sqrt{1-x}} \sim \frac{\sqrt{\pi }}{\sqrt{n}}+\frac{a_{1}\Gamma \left(\frac{3}{2}\right)}{n\sqrt{n}}+\frac{a_{2}\Gamma \left(\frac{5}{2}\right)}{n^{2}\sqrt{n}}+\ldots ,

if 11−e−t=1t+a1t+a2tt+…\dfrac {1}{\sqrt{1-e^{-t}}} = \dfrac {1}{\sqrt{t}}+a_{1}\sqrt{t}+a_{2}t\sqrt{t}+\ldots

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Problem 4797

Using the equalities:

∫0∞sin⁡u duu+m=∫0∞e−mu du1+u2,∫0∞cos⁡u duu+m=∫0∞ue−mu du1+u2, \int _{0}^{\infty } \frac{\sin u\, du}{u+m} = \int _{0}^{\infty } \frac{e^{-mu}\, du}{1+u^{2}}, \qquad \int _{0}^{\infty } \frac{\cos u\, du}{u+m} = \int _{0}^{\infty } \frac{u e^{-mu}\, du}{1+u^{2}},

prove the formula:

∫−∞∞sin⁡xx dx=cos⁡mm(1−2!m2+4!m4−…+θ(−1)ν(2ν)!m2ν)+ \int _{-\infty }^{\infty } \frac{\sin x}{x}\, dx = \frac{\cos m}{m}\left(1-\frac{2!}{m^{2}}+\frac{4!}{m^{4}}-\ldots +\theta (-1)^{\nu }\frac{(2\nu )!}{m^{2\nu }}\right)+ +sin⁡mm2[1−3!m2+5!m4−…+θ1(−1)ν(2ν+1)!m2ν], +\frac{\sin m}{m^{2}}\left[1-\frac{3!}{m^{2}}+\frac{5!}{m^{4}}-\ldots +\theta _{1}(-1)^{\nu }\frac{(2\nu +1)!}{m^{2\nu }}\right],

where θ\theta and θ1\theta_{1} are some positive proper fractions.

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Problem 4798

Find the values of the following integrals:

∫0∞dx(x4+1)2. \int _{0}^{\infty } \frac{dx}{(x^{4}+1)^{2}} .
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Problem 4799

Find the values of the following integrals:

∫0∞dx(x2+x+1)2. \int _{0}^{\infty } \frac{dx}{(x^{2}+x+1)^{2}} .
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Problem 4800

Find the values of the following integrals:

∫0∞dx(x4−x2+1)2. \int _{0}^{\infty } \frac{dx}{(x^{4}-x^{2}+1)^{2}} .
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Problem 4801

Find the values of the following integrals:

∫−∞∞x dxex−a+ea−x. \int _{-\infty }^{\infty } \frac{x\, dx}{e^{x-a}+e^{a-x}} .
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Problem 4802

Find the values of the following integrals:

∫−aadx1+φ(x)+1+φ2(x), \int _{-a}^{a} \frac{dx}{1+\varphi (x)+\sqrt{1+\varphi ^{2}(x)}},

where φ(x)\varphi (x) is an odd function.

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Problem 4803

Find the values of the following integrals:

∫0π/2arctg⁡sin⁡xsin⁡x dx. \int _{0}^{\pi /2} \frac{\arctg \sin x}{\sin x}\, dx .
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Problem 4804

Find the values of the following integrals:

∫−∞∞e−a2x2+2abx dx. \int _{-\infty }^{\infty } e^{-a^{2}x^{2}+2abx}\, dx .
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Problem 4805

Find the values of the following integrals:

∫−∞+∞cos⁡αx(x−a)2+b2 dx. \int _{-\infty }^{+\infty } \frac{\cos \alpha x}{(x-a)^{2}+b^{2}}\, dx .
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Problem 4806

Find the values of the following integrals:

∫−∞+∞sin⁡αx(x−a)2+b2 dx. \int _{-\infty }^{+\infty } \frac{\sin \alpha x}{(x-a)^{2}+b^{2}}\, dx .
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Problem 4807

Find the values of the following integrals:

∫0∞cos⁡αxcos⁡βx1+x2 dx. \int _{0}^{\infty } \frac{\cos \alpha x \cos \beta x}{1+x^{2}}\, dx .
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Problem 4808

Find the values of the following integrals:

∫0∞xe−x2sin⁡ax dx. \int _{0}^{\infty } x e^{-x^{2}} \sin ax\, dx .
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Problem 4809

Find the values of the following integrals:

∫0∞∫0∞e−x2−y2+2xycos⁡α dx dy. \int _{0}^{\infty } \int _{0}^{\infty } e^{-x^{2}-y^{2}+2xy\cos \alpha }\, dx\, dy .
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Problem 4810

Find the values of the following integrals:

∫0∞ln⁡x(1+x)4 dx. \int _{0}^{\infty } \frac{\ln x}{(1+x)^{4}}\, dx .
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Problem 4811

Find the values of the following integrals:

∫0∞x2 dx(x2−a2)2+b2x2. \int _{0}^{\infty } \frac{x^{2}\, dx}{(x^{2}-a^{2})^{2}+b^{2}x^{2}} .
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Problem 4812

Find the values of the following integrals:

∫0∞ln⁡(1+x)xλ dx. \int _{0}^{\infty } \frac{\ln (1+x)}{x^{\lambda }}\, dx .
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Problem 4813

Find the values of the following integrals:

∫−∞∞(1+x2n−1)−n+12 dx. \int _{-\infty }^{\infty } \left(1+\frac{x^{2}}{n-1}\right)^{-\frac{n+1}{2}}\, dx .
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Problem 4814

Find the values of the following integrals:

∫01dx(1−x)2x2(x−α)23. \int _{0}^{1} \frac{dx}{\sqrt[3]{(1-x)^{2} x^{2}(x-\alpha )^{2}}} .
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Problem 4815

Find the values of the following integrals:

∫01xa−xbln⁡x dx. \int _{0}^{1} \frac{x^{a}-x^{b}}{\ln x}\, dx .
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Problem 4816

Find the values of the following integrals:

∫0∞cos⁡ax−cos⁡bxx dx;a>0,b>0. \int _{0}^{\infty } \frac{\cos ax-\cos bx}{x}\, dx; \quad a > 0, \quad b > 0 .
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Problem 4817

Find the values of the following integrals:

∫0∞sin⁡axsin⁡bxx dx;a>0,b>0. \int _{0}^{\infty } \frac{\sin ax \sin bx}{x}\, dx; \quad a > 0, \quad b > 0 .
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Problem 4818

Find the values of the following integrals:

∫0∞arctg⁡ax−arctg⁡bxx dx;a>0,b>0. \int _{0}^{\infty } \frac{\arctg ax-\arctg bx}{x}\, dx; \quad a > 0, \quad b > 0 .
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Problem 4819

Find the values of the following integrals:

∫0∞sin⁡ax−asin⁡xx2 dx;a⩾0. \int _{0}^{\infty } \frac{\sin ax-a\sin x}{x^{2}}\, dx; \quad a \geqslant 0 .
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Problem 4820

Find the values of the following integrals:

∫0∞axcos⁡x−sin⁡axx2 dx;a⩾0. \int _{0}^{\infty } \frac{ax\cos x-\sin ax}{x^{2}}\, dx; \quad a \geqslant 0 .
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Problem 4821

Find the values of the following integrals:

∫0∞1−cos⁡axx2 dx. \int _{0}^{\infty } \frac{1-\cos ax}{x^{2}}\, dx .
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Problem 4822

Find the values of the following integrals:

∫0∞cos⁡ax(1+x2)2 dx. \int _{0}^{\infty } \frac{\cos ax}{(1+x^{2})^{2}}\, dx .
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Problem 4823

Find the values of the following integrals:

∫0∞sin⁡2xx2+a2 dx. \int _{0}^{\infty } \frac{\sin ^{2} x}{x^{2}+a^{2}}\, dx .
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Problem 4824

Find the values of the following integrals:

∫0∞sin⁡4axx4 dx. \int _{0}^{\infty } \frac{\sin ^{4} ax}{x^{4}}\, dx .
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Problem 4825

Find the values of the following integrals:

∫0∞sin⁡axsin⁡bxx2 dx;a⩾b>0. \int _{0}^{\infty } \frac{\sin ax \sin bx}{x^{2}}\, dx; \quad a \geqslant b > 0 .
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Problem 4826

Find the values of the following integrals:

∫0∞sin⁡2xcos⁡axx2 dx;a>0. \int _{0}^{\infty } \frac{\sin ^{2} x \cos ax}{x^{2}}\, dx; \quad a > 0 .
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Problem 4827

Find the values of the following integrals:

∫0∞sin⁡3xcos⁡axx dx;a>0. \int _{0}^{\infty } \frac{\sin ^{3} x \cos ax}{x}\, dx; \quad a > 0 .
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Problem 4828

Find the values of the following integrals:

∫0∞sin⁡axsin⁡2xx3 dx. \int _{0}^{\infty } \frac{\sin ax \sin ^{2} x}{x^{3}}\, dx .
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Problem 4829

Find the values of the following integrals:

∫0∞x−sin⁡xx3 dx. \int _{0}^{\infty } \frac{x-\sin x}{x^{3}}\, dx .
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Problem 4830

Find the values of the following integrals:

∫0∞sin⁡x−xcos⁡xx3 dx. \int _{0}^{\infty } \frac{\sin x-x\cos x}{x^{3}}\, dx .
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Problem 4831

Find the values of the following integrals:

∫−∞∞cos⁡mxex+e−x dx. \int _{-\infty }^{\infty } \frac{\cos mx}{e^{x}+e^{-x}}\, dx .
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Problem 4832

Find the values of the following integrals:

∫0∞e−x2xsin⁡2x−sin⁡2xx2 dx. \int _{0}^{\infty } e^{-x^{2}} \frac{x\sin 2x-\sin ^{2}x}{x^{2}}\, dx .
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Problem 4833

Find the values of the following integrals:

∫0∞xn−1ln⁡x(x2+a2)n dx. \int _{0}^{\infty } \frac{x^{n-1}\ln x}{(x^{2}+a^{2})^{n}}\, dx .
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Problem 4834

Find the values of the following integrals:

∫0∞xsin⁡ax(1+x2)2 dx. \int _{0}^{\infty } \frac{x\sin ax}{(1+x^{2})^{2}}\, dx .
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Problem 4835

Find the values of the following integrals:

∫0∞x2−a2x2+a2sin⁡xx dx. \int _{0}^{\infty } \frac{x^{2}-a^{2}}{x^{2}+a^{2}} \frac{\sin x}{x}\, dx .
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Problem 4836

Find the values of the following integrals:

v.p.∫0π/2dxa−bcos⁡x,0<a<b. \text{v.p.} \int _{0}^{\pi /2} \frac{dx}{a-b\cos x}, \quad 0 < a < b .
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Problem 4837

Find the values of the following integrals:

v.p.∫0∞dx1−x3. \text{v.p.} \int _{0}^{\infty } \frac{dx}{1-x^{3}} .
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Problem 4838

Find the values of the following integrals:

∫0πxsin⁡x1+cos⁡2x dx. \int _{0}^{\pi } \frac{x\sin x}{1+\cos ^{2}x}\, dx .
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Problem 4839

Find the values of the following integrals:

∫02πx dx1−acos⁡x,for ∣a∣<1. \int _{0}^{2\pi } \frac{x\, dx}{1-a\cos x}, \quad \text{for } \left|a\right| < 1 .
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Problem 4840

Find the values of the following integrals:

v.p.∫02πx dx1−acos⁡x,for ∣a∣>1. \text{v.p.} \int _{0}^{2\pi } \frac{x\, dx}{1-a\cos x}, \quad \text{for } \left|a\right| > 1 .
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Problem 4841

Find the values of the following integrals:

∫0∞e−x dxsh⁡2x. \int _{0}^{\infty } \frac{e^{-x}\, dx}{\sqrt{\operatorname {sh}2x}} .
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Problem 4842

Find the values of the following integrals:

∫0∞e−αx dxsh⁡2x. \int _{0}^{\infty } \frac{e^{-\alpha x}\, dx}{\sqrt{\operatorname {sh}2x}} .
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Problem 4843

Find the values of the following integrals:

∫0πdx1−2sin⁡2αcos⁡x+cos⁡2x;−π4<α<π4. \int _{0}^{\pi } \frac{dx}{1-2\sin 2\alpha \cos x+\cos ^{2} x}; \quad -\frac{\pi }{4} < \alpha < \frac{\pi }{4} .
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Problem 4844

Find the values of the following integrals:

∫0∞e−x(1−cos⁡ax)x dx. \int _{0}^{\infty } \frac{e^{-x}(1-\cos ax)}{x}\, dx .
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Problem 4845

Find the value of the integral ∫0∞J0(x)e−bx dx\int_{0}^{\infty } J_{0}(x) e^{-bx}\, dx, where

J0(x)=2π∫0π/2cos⁡(xcos⁡φ) dφ. J_{0}(x) = \frac{2}{\pi } \int _{0}^{\pi /2} \cos (x\cos \varphi )\, d\varphi .
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Problem 4846

Prove that ∫0π/2sin⁡pxsin⁡xcos⁡p−1x dx=π2\int_{0}^{\pi /2} \dfrac {\sin px}{\sin x} \cos^{p-1} x\, dx = \dfrac {\pi }{2} for p>0p > 0. (Liouville.)

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Problem 4847

Prove the Ramanujan identity:

α∫0∞e−x2 dxch⁡αx=β∫0∞e−x2 dxch⁡βx;αβ=π. \sqrt{\alpha } \int _{0}^{\infty } \frac{e^{-x^{2}}\, dx}{\operatorname {ch}\alpha x} = \sqrt{\beta } \int _{0}^{\infty } \frac{e^{-x^{2}}\, dx}{\operatorname {ch}\beta x}; \quad \alpha \beta = \pi .
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Problem 4848

Prove that as n→∞n \rightarrow \infty the absolute value of the integral

∫0nx(1−x21)(1−x24)…(1−x2n2) dx→∞. \int _{0}^{n} x\left(1-\frac{x^{2}}{1}\right)\left(1-\frac{x^{2}}{4}\right)\ldots \left(1-\frac{x^{2}}{n^{2}}\right)\, dx \rightarrow \infty .
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Problem 4849

Prove the equality:

∫0∞f(x+ax) dx=∫0∞f(x2+4a) dx,a>0, \int _{0}^{\infty } f\left(x+\frac{a}{x}\right)\, dx = \int _{0}^{\infty } f\left(\sqrt{x^{2}+4a}\right)\, dx, \quad a > 0,

assuming that both integrals make sense.

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Problem 4850

Using the Fresnel integrals and the preceding result, prove the equality:

∫0∞cos⁡(x2+a2x2) dx=12π2(cos⁡2a−sin⁡2a). \int _{0}^{\infty } \cos \left(x^{2}+\frac{a^{2}}{x^{2}}\right)\, dx = \frac{1}{2}\sqrt{\frac{\pi }{2}}(\cos 2a-\sin 2a) .
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Problem 4851

Using the substitution x−ab−x=αβete−t\dfrac {x-a}{b-x} = \dfrac {\alpha }{\beta }\dfrac {e^{t}}{e^{-t}}, prove the equality:

∫ab((x−a)(b−x))−32e(−α2x−a+β2b−x) dx= \int _{a}^{b} \left((x-a)(b-x)\right)^{-\frac{3}{2}} e^{\left(-\frac{\alpha ^{2}}{x-a}+\frac{\beta ^{2}}{b-x}\right)}\, dx = =πα+βαβ(b−a)−32e−(α+β)2b−a. = \sqrt{\pi }\frac{\alpha +\beta }{\alpha \beta }(b-a)^{-\frac{3}{2}} e^{-\frac{(\alpha +\beta )^{2}}{b-a}} .
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Problem 4852

Using the identity

4cos⁡2x=(1+e2xi)(1+e−2xi), 4\cos ^{2}x = (1+e^{2xi})(1+e^{-2xi}),

prove the equalities, for 0<x<π20 < x < \dfrac {\pi }{2},

∫0xln⁡cos⁡x dx=−xln⁡2−12∑ν=1∞(−1)νν2sin⁡2νx, \int _{0}^{x} \ln \cos x\, dx = -x\ln 2-\frac{1}{2}\sum _{\nu =1}^{\infty } \frac{(-1)^{\nu }}{\nu ^{2}}\sin 2\nu x, ∫0xln⁡sin⁡x dx=−xln⁡2−12∑ν=1∞1ν2sin⁡2νx. \int _{0}^{x} \ln \sin x\, dx = -x\ln 2-\frac{1}{2}\sum _{\nu =1}^{\infty } \frac{1}{\nu ^{2}}\sin 2\nu x .

(N.I. Lobachevsky.)

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Problem 4853

Expanding in powers of aa, where ∣a∣<1\left|a\right| < 1, prove the equality:

∫0πsin⁡nx dx(1+acos⁡x)n+1=2nΓ(n+12)(1−a2)n+12. \int _{0}^{\pi } \frac{\sin ^{n} x\, dx}{(1+a\cos x)^{n+1}} = \frac{2^{n}\Gamma \left(\frac{n+1}{2}\right)}{(1-a^{2})^{\frac{n+1}{2}}} .

(N.I. Lobachevsky.)

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Problem 4854

Verify the equality:

∫0∞(ex−e−x)x dxe2x+e−2x+2cos⁡2α=πa4sin⁡α;0<α<π2. \int _{0}^{\infty } \frac{(e^{x}-e^{-x})x\, dx}{e^{2x}+e^{-2x}+2\cos 2\alpha } = \frac{\pi a}{4\sin \alpha }; \quad 0 < \alpha < \frac{\pi }{2} .

(N.I. Lobachevsky.)

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Problem 4855

Verify the equality:

∫0∞(ex−e−x)x dxe2x+e−2x+2cos⁡2α=π(π−α)4sin⁡α;π2<α<π. \int _{0}^{\infty } \frac{(e^{x}-e^{-x})x\, dx}{e^{2x}+e^{-2x}+2\cos 2\alpha } = \frac{\pi (\pi -\alpha )}{4\sin \alpha }; \quad \frac{\pi }{2} < \alpha < \pi .

(M.V. Ostrogradsky.)

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Problem 4856

Verify the equality:

∫0∞e−n2+x2cos⁡αx dx=11+α2∫0∞e−1+α2n2+x2 dx. \int _{0}^{\infty } e^{-\sqrt{n^{2}+x^{2}}} \cos \alpha x\, dx = \frac{1}{\sqrt{1+\alpha ^{2}}} \int _{0}^{\infty } e^{-\sqrt{1+\alpha ^{2}}\sqrt{n^{2}+x^{2}}}\, dx .

(N.I. Lobachevsky.)

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Section 11.9Fourier Series and Related Questions[98/52%]
Problem 4857

Expand in the simplest Fourier series the function f(x)f(x), defined on the interval (−π,π)(-\pi ,\pi ) by the conditions:

f(x)={π4for 0<x<π;−π4for −π<x<0. f(x) = \begin{cases} \dfrac {\pi }{4} & \text{for } 0 < x < \pi ; \\ -\dfrac {\pi }{4} & \text{for } -\pi < x < 0 . \end{cases}
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Problem 4858

Expand in the simplest Fourier series the function f(x)f(x), defined on the interval (−π,π)(-\pi ,\pi ) by the conditions:

f(x)={0for −π<x<0;1for 0<x<π. f(x) = \begin{cases} 0 & \text{for } -\pi < x < 0; \\ 1 & \text{for } 0 < x < \pi . \end{cases}
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Problem 4859

Expand in the simplest Fourier series the function f(x)=∣x∣f(x) = \left|x\right| for −π<x<π-\pi < x < \pi.

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Problem 4860

Expand in the simplest Fourier series the function f(x)=xf(x) = x for −π<x<π-\pi < x < \pi.

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Problem 4861

Expand in the simplest Fourier series the function f(x)f(x), defined on the interval (−π,π)(-\pi ,\pi ) by the conditions:

f(x)={sin⁡12x,0<x<π;1+cos⁡12x,−π<x<0. f(x) = \begin{cases} \sin \dfrac {1}{2}x, & 0 < x < \pi ; \\ 1+\cos \dfrac {1}{2}x, & -\pi < x < 0 . \end{cases}
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Problem 4862

Expand in the simplest Fourier series the function f(x)f(x), defined on the interval (−π,π)(-\pi ,\pi ) by the conditions:

f(x)={π−x2x,0<x<π;−π+x2x,−π<x<0. f(x) = \begin{cases} \dfrac {\pi -x}{2}x, & 0 < x < \pi ; \\ -\dfrac {\pi +x}{2}x, & -\pi < x < 0 . \end{cases}
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Problem 4863

Expand in the simplest Fourier series the function f(x)=x2f(x) = x^{2} for −π<x<π-\pi < x < \pi.

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Problem 4864

Expand in the simplest Fourier series the function f(x)f(x), defined on the interval (−π,π)(-\pi ,\pi ) by the conditions:

f(x)={cos⁡x,0<x<π;−cos⁡x,−π<x<0. f(x) = \begin{cases} \cos x, & 0 < x < \pi ; \\ -\cos x, & -\pi < x < 0 . \end{cases}
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Problem 4865

Expand in the simplest Fourier series the function f(x)=∣sin⁡x∣f(x) = \left|\sin x\right| for −π<x<π-\pi < x < \pi.

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Problem 4866

Expand in the simplest Fourier series the function f(x)=cos⁡axf(x) = \cos ax for −π<x<π-\pi < x < \pi, aa not an integer.

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Problem 4867

Expand in the simplest Fourier series the function f(x)=sin⁡axf(x) = \sin ax for −π<x<π-\pi < x < \pi, aa not an integer.

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Problem 4868

Expand in the simplest Fourier series the function f(x)f(x), defined on the interval (−π,π)(-\pi ,\pi ) by the conditions: f(x)=0f(x) = 0 for −π<x<0-\pi < x < 0; f(x)=sin⁡2xf(x) = \sin^{2} x for 0<x<π0 < x < \pi.

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Problem 4869

Expand in a Fourier series the periodic function: f(x)={x}f(x) = \left\{ x\right\}, the fractional part of xx, i.e. the difference x−[x]x-[x], where [x][x] is the integer part of xx.

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Problem 4870

Expand in a Fourier series the periodic function: f(x)=(x)f(x) = (x), the distance from xx to the nearest integer.

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Problem 4871

Expand in a Fourier series the periodic function: f(x)=xf(x) = x, 11, or 3−x3-x, according as xx lies in the interval (0,1)(0,1), (1,2)(1,2), or (2,3)(2,3), assuming that the period of f(x)f(x) equals 33.

If the function is not given as periodic and one seeks an expansion valid only on a given interval, then it can be expanded in a Fourier series with period equal to the length of the given interval, or with any larger period, but in the latter case the function must be extended in some way (as long as the Dirichlet conditions are satisfied). This can be used to obtain an expansion in cosines alone or in sines alone, or to restrict oneself to only the sines or cosines of odd multiples of the angle.

In the problems that follow, it will be assumed that the period is to be chosen as the smallest one for which the requirements of the problem are satisfied.

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Problem 4872

On the interval (−h,h)(-h,h), expand the function exe^{x}.

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Problem 4873

Expand ch⁡ax\operatorname {ch}ax on the interval (−π,π)(-\pi ,\pi ).

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Problem 4874

Expand sh⁡ax\operatorname {sh}ax on the interval (−π,π)(-\pi ,\pi ).

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Problem 4875

On the interval (0,π)(0,\pi ), expand x3x^{3} in cosines.

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Problem 4876

On the interval (0,π)(0,\pi ), expand x3x^{3} in sines.

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Problem 4877

On the interval (0,π)(0,\pi ), expand sin⁡x\sin x in cosines.

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Problem 4878

The function f(x)=x2(a−x)f(x) = x^{2}(a-x) is given on the interval (0,a)(0,a). Expand it in a Fourier series with period aa.

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Problem 4879

The function f(x)=x2(a−x)f(x) = x^{2}(a-x) is given on the interval (0,a)(0,a). Expand it in a Fourier series as an even function.

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Problem 4880

The function f(x)=x2(a−x)f(x) = x^{2}(a-x) is given on the interval (0,a)(0,a). Expand it in a Fourier series as an odd function.

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Problem 4881

The function f(x)=xf(x) = x is given on the interval (0,a)(0,a). Expand it in a Fourier series with period aa.

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Problem 4882

The function f(x)=xf(x) = x is given on the interval (0,a)(0,a). Expand it in a Fourier series as an even function.

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Problem 4883

The function f(x)=xf(x) = x is given on the interval (0,a)(0,a). Expand it in a Fourier series as an odd function.

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Problem 4884

The function f(x)=xf(x) = x is given on the interval (0,a)(0,a). Expand it in a Fourier series as an odd function, in sines of odd multiples of the angle.

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Problem 4885

Functions rational in cos⁡x\cos x and sin⁡x\sin x can be expanded in a Fourier series without using the Fourier formulas. To do this we write:

cos⁡x=t2+12t,sin⁡x=t2−12ti, \cos x = \frac{t^{2}+1}{2t}, \qquad \sin x = \frac{t^{2}-1}{2ti},

where t=exit = e^{xi}. The problem then reduces to expanding the function in positive and negative powers of tt. Passing back from tt to xx, we obtain the desired Fourier series expansion.

By the method just described and similar ones, obtain the following expansions:

1−a21−2acos⁡x+a2=1+2∑n=1∞ancos⁡nx;∣a∣<1. \frac{1-a^{2}}{1-2a\cos x+a^{2}} = 1+2\sum _{n=1}^{\infty } a^{n}\cos nx; \quad \left|a\right| < 1 .
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Problem 4886

By the method just described, obtain the following expansion:

1−acos⁡x1−2acos⁡x+a2=∑n=0∞ancos⁡nx;∣a∣<1. \frac{1-a\cos x}{1-2a\cos x+a^{2}} = \sum _{n=0}^{\infty } a^{n}\cos nx; \quad \left|a\right| < 1 .
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Problem 4887

By the method just described, obtain the following expansion:

asin⁡x1−2acos⁡x+a2=∑n=0∞ansin⁡nx;∣a∣<1. \frac{a\sin x}{1-2a\cos x+a^{2}} = \sum _{n=0}^{\infty } a^{n}\sin nx; \quad \left|a\right| < 1 .
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Problem 4888

By the method just described, obtain the following expansion:

ln⁡(1−2acos⁡x+a2)=−2∑n=1∞anncos⁡nx;∣a∣<1. \ln (1-2a\cos x+a^{2}) = -2\sum _{n=1}^{\infty } \frac{a^{n}}{n}\cos nx; \quad \left|a\right| < 1 .
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Problem 4889

By the method just described, obtain the following expansion:

ln⁡∣2sin⁡x2∣−1=cos⁡x+cos⁡2x2+cos⁡3x3+… \ln \left|2\sin \frac{x}{2}\right|^{-1} = \cos x+\frac{\cos 2x}{2}+\frac{\cos 3x}{3}+\ldots
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Problem 4890

The function f(x)f(x), given on the interval (0,π2)\left(0,\dfrac {\pi }{2}\right), is continued onto the interval (π2,π)\left(\dfrac {\pi }{2},\pi \right), so that f(π2+x)=−f(π2−x)f\left(\dfrac {\pi }{2}+x\right) = -f\left(\dfrac {\pi }{2}-x\right); prove that for 0<x<π0 < x < \pi the equality holds

f(x)=∑ancos⁡(2n+1)x, f(x) = \sum a_{n}\cos (2n+1)x,

where

an=4π∫0π/2f(x)cos⁡(2n+1)x dx. a_{n} = \frac{4}{\pi } \int _{0}^{\pi /2} f(x)\cos (2n+1)x\, dx .
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Problem 4891

How should the function f(x)f(x), given on the interval (0,π2)\left(0,\dfrac {\pi }{2}\right), be continued onto the interval (π2,π)\left(\dfrac {\pi }{2},\pi \right) so that for 0<x<π0 < x < \pi

f(x)=∑bnsin⁡(2n+1)x f(x) = \sum b_{n}\sin (2n+1)x

and how can the coefficients be computed?

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Problem 4892

Prove, for 0<x<π0 < x < \pi, the following equalities:

cos⁡2x1⋅3+cos⁡4x3⋅5+cos⁡6x5⋅7+…=12−π4sin⁡x. \frac{\cos 2x}{1 \cdot 3}+\frac{\cos 4x}{3 \cdot 5}+\frac{\cos 6x}{5 \cdot 7}+\ldots = \frac{1}{2}-\frac{\pi }{4}\sin x .
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Problem 4893

Prove, for 0<x<π0 < x < \pi, the following equalities:

sin⁡x13+sin⁡3x33+sin⁡5x53+…=πx8(π−x). \frac{\sin x}{1^{3}}+\frac{\sin 3x}{3^{3}}+\frac{\sin 5x}{5^{3}}+\ldots = \frac{\pi x}{8}(\pi -x) .
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Problem 4894

Prove the equality:

∑n=1∞sin⁡(2n+2)θn(n+1)= \sum _{n=1}^{\infty } \frac{\sin (2n+2)\theta }{n(n+1)} = =sin⁡2θ−(π−2θ)sin⁡2θ−sin⁡θcos⁡θln⁡4sin⁡2θ. = \sin 2\theta -(\pi -2\theta )\sin ^{2}\theta -\sin \theta \cos \theta \ln 4\sin ^{2}\theta .
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Problem 4895

Prove the equality:

∑n=1∞cos⁡(2n+1)θn2(n+1)2= \sum _{n=1}^{\infty } \frac{\cos (2n+1)\theta }{n^{2}(n+1)^{2}} = =2(π−2θ)sin⁡θ+(π23−2πθ+2θ2−3)cos⁡θ. = 2(\pi -2\theta )\sin \theta +\left(\frac{\pi ^{2}}{3}-2\pi \theta +2\theta ^{2}-3\right)\cos \theta .
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Problem 4896

Prove the correctness of the expansion:

cos⁡3x1⋅3⋅5−cos⁡5x3⋅5⋅7+cos⁡7x5⋅7⋅9−…=π8cos⁡2x−13cos⁡x \frac{\cos 3x}{1 \cdot 3 \cdot 5}-\frac{\cos 5x}{3 \cdot 5 \cdot 7}+\frac{\cos 7x}{5 \cdot 7 \cdot 9}-\ldots = \frac{\pi }{8}\cos ^{2}x-\frac{1}{3}\cos x

on the interval 0<x<π20 < x < \dfrac {\pi }{2}.

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Problem 4897

Prove the equalities:

∑n=−∞∞e−πa(n+x)2=1a∑n=−∞∞e−πn2acos⁡2πnx,a>0. \sum _{n=-\infty }^{\infty } e^{-\pi a(n+x)^{2}} = \frac{1}{\sqrt{a}} \sum _{n=-\infty }^{\infty } e^{-\frac{\pi n^{2}}{a}} \cos 2\pi nx, \quad a > 0 .
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Problem 4898

Prove the equalities:

12+∑n=1∞e−πn2x=1x(12+∑n=1∞e−πn2x),x>0. \frac{1}{2}+\sum _{n=1}^{\infty } e^{-\pi n^{2} x} = \frac{1}{\sqrt{x}}\left(\frac{1}{2}+\sum _{n=1}^{\infty } e^{-\frac{\pi n^{2}}{x}}\right), \quad x > 0 .
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Problem 4899

Prove the equalities:

lim⁡x→1−01−x (x+x4+x9+…)=12π. \lim _{x \rightarrow 1-0} \sqrt{1-x}\, (x+x^{4}+x^{9}+\ldots ) = \frac{1}{2}\sqrt{\pi } .
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Problem 4900

Prove the equalities:

lim⁡x→1−0(x−x4+x9−x16+…)=12. \lim _{x \rightarrow 1-0} (x-x^{4}+x^{9}-x^{16}+\ldots ) = \frac{1}{2} .
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Problem 4901

Prove the equality (Poisson's formula):

∑n=abf(n)= \sum _{n=a}^{b} f(n) = =f(a)+f(b)2−∫abf(x) dx+2∑n=1∞∫abf(x)cos⁡2πnx dx. = \frac{f(a)+f(b)}{2}-\int _{a}^{b} f(x)\, dx+2\sum _{n=1}^{\infty } \int _{a}^{b} f(x)\cos 2\pi nx\, dx .
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Problem 4902

The modulus of continuity ω(h)\omega (h) is the exact upper bound of the oscillation of the function f(x)f(x) on intervals of length hh, i.e. of the numbers ∣f(x2)−f(x1)∣\left|f(x_{2})-f(x_{1})\right|, where ∣x2−x1∣<h\left|x_{2}-x_{1}\right| < h. Prove that for a continuous function f(x)f(x) the integrals

∫abf(x)cos⁡nx dxand∫abf(x)sin⁡nx dx \int _{a}^{b} f(x)\cos nx\, dx \quad \text{and} \quad \int _{a}^{b} f(x)\sin nx\, dx

are, in absolute value, no greater than Cω(πn)C\omega \left(\dfrac {\pi }{n}\right), where CC is a corresponding constant.

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Problem 4903

Prove the equality:

∫0π/2∣sin⁡mx∣sin⁡x dx=2πln⁡m+O(1), \int _{0}^{\pi /2} \frac{\left|\sin mx\right|}{\sin x}\, dx = \frac{2}{\pi }\ln m+O(1),

where O(1)O(1), as m→∞m \rightarrow \infty, is a bounded quantity.

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Problem 4904

Prove Lebesgue's inequality:

∣a02+∑n=1m(ancos⁡nx+bnsin⁡nx)∣<CMln⁡m. \left|\frac{a_{0}}{2}+\sum _{n=1}^{m} (a_{n}\cos nx+b_{n}\sin nx)\right| < CM\ln m .

Here ana_{n} and bnb_{n} are the Fourier coefficients of the function f(x)f(x), and MM is an upper bound for ∣f(x)∣\left|f(x)\right|.

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Problem 4905

Prove that for 0<x<π0 < x < \pi the equality holds:

∑ν=1nsin⁡νxν=−x2+∫0x/2sin⁡(2n+1)tt dt+ \sum _{\nu =1}^{n} \frac{\sin \nu x}{\nu } = -\frac{x}{2}+\int _{0}^{x/2} \frac{\sin (2n+1)t}{t}\, dt+ +∫0x/2(1sin⁡t−1t)sin⁡(2n+1)t dt. +\int _{0}^{x/2} \left(\frac{1}{\sin t}-\frac{1}{t}\right) \sin (2n+1)t\, dt .
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Problem 4906

Prove that for any real xx and integer n>0n > 0 the equality holds

∑ν=1nsin⁡νxν=2πθ;∣θ∣<1. \sum _{\nu =1}^{n} \frac{\sin \nu x}{\nu } = 2\pi \theta ; \quad \left|\theta \right| < 1 .
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Problem 4907

For 0<x<π0 < x < \pi, prove the equality:

∑ν=1nsin⁡(2ν−1)x2ν−1= \sum _{\nu =1}^{n} \frac{\sin (2\nu -1)x}{2\nu -1} = =12∫0xsin⁡2ntt dt+12∫0x(1sin⁡t−1t)sin⁡2nt dt. = \frac{1}{2}\int _{0}^{x} \frac{\sin 2nt}{t}\, dt+\frac{1}{2}\int _{0}^{x} \left(\frac{1}{\sin t}-\frac{1}{t}\right) \sin 2nt\, dt .
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Problem 4908

For any real xx and integer n>0n > 0, prove the equality:

∑ν=1nsin⁡(2ν−1)x2ν−1=32θ;∣θ∣<1. \sum _{\nu =1}^{n} \frac{\sin (2\nu -1)x}{2\nu -1} = \frac{3}{2}\theta ; \quad \left|\theta \right| < 1 .
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Problem 4909

Prove Fejér's inequality:

∣cos⁡xn+cos⁡2xn−1+…+cos⁡nx1−cos⁡(n+1)x1−cos⁡(n+2)x2−…−cos⁡2nxn∣<10. \left|\frac{\cos x}{n}+\frac{\cos 2x}{n-1}+\ldots +\frac{\cos nx}{1}-\frac{\cos (n+1)x}{1}-\frac{\cos (n+2)x}{2}-\ldots -\frac{\cos 2nx}{n}\right| < 10 .
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Problem 4910

The Fejér sum σn\sigma_{n} is the arithmetic mean of the first nn Fourier sums of the function f(x)f(x):

nσn=s0+s1+…+sn−1, n\sigma _{n} = s_{0}+s_{1}+\ldots +s_{n-1},

where for m⩾1m \geqslant 1

sm=a02+∑ν=1m(aνcos⁡νx+bνsin⁡νx);s0=a02, s_{m} = \frac{a_{0}}{2}+\sum _{\nu =1}^{m} (a_{\nu }\cos \nu x+b_{\nu }\sin \nu x); \quad s_{0} = \frac{a_{0}}{2},

aνa_{\nu }, bνb_{\nu } being the Fourier coefficients of the function f(x)f(x). Prove that ∣σn(x)∣<M\left|\sigma_{n}(x)\right| < M, if ∣f(x)∣<M\left|f(x)\right| < M throughout the period.

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Problem 4911

Prove that if f(x)f(x) is a continuous function on the interval [−π,π]\left[-\pi ,\pi \right], l⩽f(x)⩽Ll \leqslant f(x) \leqslant L, and for every mm, ∣mam∣<A\left|ma_{m}\right| < A, ∣mbm∣<B\left|mb_{m}\right| < B, then

l−(A+B)⩽sn(x)⩽L+(A+B); l-(A+B) \leqslant s_{n}(x) \leqslant L+(A+B);

where ama_{m}, bmb_{m} are the Fourier coefficients, sn(x)s_{n}(x) is the nn-th partial sum of the Fourier series of f(x)f(x).

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Problem 4912

Prove that for every continuous function f(x)f(x) on the interval [−π,π]\left[-\pi ,\pi \right], σn(x)→f(x)\sigma_{n}(x) \rightarrow f(x) as n→∞n \rightarrow \infty.

?
Problem 4913

Prove that for a periodic function f(x)f(x) satisfying the Cauchy--Lipschitz condition

∣f(x2)−f(x1)∣<C∣x2−x1∣, \left|f(x_{2})-f(x_{1})\right| < C\left|x_{2}-x_{1}\right|,

the inequality ∣σn−f(x)∣<Aln⁡nn\left|\sigma_{n}-f(x)\right| < \dfrac {A\ln n}{n} holds, where AA is a constant.

?
Problem 4914

Taking f(x)f(x), wherever it occurs, to be a continuous function with period 2π2\pi, prove the following formulas and assertions:

The integral 2π⋅(2n)!!(2n−1)!!∫−ππf(x+u)cos⁡2nu2 du\dfrac {2}{\pi } \cdot \dfrac {(2n)!!}{(2n-1)!!} \int_{-\pi }^{\pi } f(x+u) \cos^{2n}\dfrac {u}{2}\, du is a trigonometric polynomial that tends to f(x)f(x) as n→∞n \rightarrow \infty.

?
Problem 4915

Prove that

∫0π/2Fm(u) du>π3m−111m2;2π∫0π/2uFm(u) du<cm2, \int _{0}^{\pi /2} F_{m}(u)\, du > \frac{\pi }{3m}-\frac{1}{11m^{2}}; \qquad 2\pi \int _{0}^{\pi /2} u F_{m}(u)\, du < \frac{c}{m^{2}},

where Fm(u)=(sin⁡mumsin⁡u)4F_{m}(u) = \left(\dfrac {\sin mu}{m\sin u}\right)^{4}, and cc is some constant.

∫0π/2Fm(u) du=π3m+π6m3=12Am. \int _{0}^{\pi /2} F_{m}(u)\, du = \frac{\pi }{3m}+\frac{\pi }{6m^{3}} = \frac{1}{2A_{m}} .
?
Problem 4916

The quantity Tm(x)T_{m}(x), where

Tm(x)=Am∫−π/2π/2f(x+2u)Fm(u) du, T_{m}(x) = A_{m} \int _{-\pi /2}^{\pi /2} f(x+2u) F_{m}(u)\, du,

(Fm(u)F_{m}(u) and AmA_{m} are defined in problem 4915) is a trigonometric polynomial of order 2m−22m-2.

?
Problem 4917

Prove the equality:

f(x)−Tm(x)=Am∫−π/2π/2(f(x)−f(x+2u))Fm(u) du. f(x)-T_{m}(x) = A_{m} \int _{-\pi /2}^{\pi /2} \left(f(x)-f(x+2u)\right) F_{m}(u)\, du .

For the value of AmA_{m}, see problem 4915.

?
Problem 4918

If f(x)f(x) satisfies the condition

∣f(x1)−f(x2)∣<C∣x2−x1∣, \left|f(x_{1})-f(x_{2})\right| < C\left|x_{2}-x_{1}\right|,

then Jackson's inequality holds:

∣f(x)−Tm(x)∣<Mm, \left|f(x)-T_{m}(x)\right| < \frac{M}{m},

where MM is a corresponding constant.

?
Problem 4919

Let f(x)f(x) be a function integrable on the interval (0,2π)(0,2\pi ), let ana_{n} and bnb_{n} be its Fourier coefficients, let Sm=a02+∑n=1m(ancos⁡nx+bnsin⁡nx)S_{m} = \dfrac {a_{0}}{2}+\sum_{n=1}^{m} (a_{n}\cos nx+b_{n}\sin nx) be its Fourier sum, and let Pm(x)P_{m}(x) be any trigonometric polynomial of order mm, different from Sm(x)S_{m}(x). Prove the inequality:

∫02π(f(x)−Sm(x))2 dx<∫02π(f(x)−Pm(x))2 dx. \int _{0}^{2\pi } \left(f(x)-S_{m}(x)\right)^{2}\, dx < \int _{0}^{2\pi } \left(f(x)-P_{m}(x)\right)^{2}\, dx .
?
Problem 4920

Prove Bessel's inequality:

a022+∑ν=1∞(aν2+bν2)⩽1π∫02π∣f(x)∣2 dx. \frac{a_{0}^{2}}{2}+\sum _{\nu =1}^{\infty } (a_{\nu }^{2}+b_{\nu }^{2}) \leqslant \frac{1}{\pi } \int _{0}^{2\pi } \left|f(x)\right|^{2}\, dx .
?
Problem 4921

Starting from Bessel's inequality and the inequality 2ab⩽a2+b22ab \leqslant a^{2}+b^{2}, prove the convergence of the series ∑ν=1∞(aν2+bν2)\sum_{\nu =1}^{\infty } (a_{\nu }^{2}+b_{\nu }^{2}) and ∑ν=1∞∣aν∣+∣bν∣να\sum_{\nu =1}^{\infty } \dfrac {\left|a_{\nu }\right|+\left|b_{\nu }\right|}{\nu^{\alpha }}, where α>12\alpha > \dfrac {1}{2}.

?
Problem 4922

Prove the inequality:

∫ab∣f(x)−φ(x)∣2 dx<2∫ab∣f(x)−ψ(x)∣2 dx+2∫ab∣ψ(x)−φ(x)∣2 dx. \int _{a}^{b} \left|f(x)-\varphi (x)\right|^{2}\, dx < 2\int _{a}^{b} \left|f(x)-\psi (x)\right|^{2}\, dx+2\int _{a}^{b} \left|\psi (x)-\varphi (x)\right|^{2}\, dx .
?
Problem 4923

Let f(x)f(x) be a function integrable on (0,2π)(0,2\pi ), let ε\varepsilon be an arbitrary positive constant, let φ(x)\varphi (x) be a continuous function such that ∫02π∣f(x)−φ(x)∣2 dx<ε\int_{0}^{2\pi } \left|f(x)-\varphi (x)\right|^{2}\, dx < \varepsilon, and let σn(x)\sigma_{n}(x) be the Fejér sum (see problem 4910) for φ(x)\varphi (x), differing from φ(x)\varphi (x) by less than ε2π\sqrt{\dfrac {\varepsilon }{2\pi }}. Prove the inequality:

∫02π∣f(x)−σn(x)∣2 dx<4ε. \int _{0}^{2\pi } \left|f(x)-\sigma _{n}(x)\right|^{2}\, dx < 4\varepsilon .
?
Problem 4924

Using the preceding results, prove A.M. Lyapunov's theorem (the completeness theorem):

a022+∑ν=1∞(aν2+bν2)=1π∫02πf2(x) dx, \frac{a_{0}^{2}}{2}+\sum _{\nu =1}^{\infty } (a_{\nu }^{2}+b_{\nu }^{2}) = \frac{1}{\pi } \int _{0}^{2\pi } f^{2}(x)\, dx,

valid for any integrable function f(x)f(x).

?
Problem 4925

Prove the equality:

1π∫02πφ(x)ψ(x) dx=a0α02+∑ν=1∞(aναν+bνβν), \frac{1}{\pi } \int _{0}^{2\pi } \varphi (x)\psi (x)\, dx = \frac{a_{0}\alpha _{0}}{2}+\sum _{\nu =1}^{\infty } (a_{\nu }\alpha _{\nu }+b_{\nu }\beta _{\nu }),

where φ(x)\varphi (x) and ψ(x)\psi (x) are integrable on the interval (0,2π)(0,2\pi ), and aνa_{\nu }, bνb_{\nu }, αν\alpha_{\nu }, βν\beta_{\nu } are the Fourier coefficients of these functions.

?
Problem 4926

Prove V.A. Steklov's theorem:

12h∫x−hx+hf(u) du=a02+∑ν=1∞(aνcos⁡νx+bνsin⁡νx)sin⁡νhνh. \frac{1}{2h} \int _{x-h}^{x+h} f(u)\, du = \frac{a_{0}}{2}+\sum _{\nu =1}^{\infty } (a_{\nu }\cos \nu x+b_{\nu }\sin \nu x)\frac{\sin \nu h}{\nu h} .
?
Problem 4927

Prove V.A. Steklov's inequalities:

?
(a)

∫02π∣f′(x)∣2 dx⩾∫02πf2(x) dx\displaystyle \int_{0}^{2\pi } \left|f'(x)\right|^{2}\, dx \geqslant \int_{0}^{2\pi } f^{2}(x)\, dx, if ∫0πf(x) dx=0\displaystyle \int_{0}^{\pi } f(x)\, dx = 0;

(b)

∫0π∣f′(x)∣2 dx⩾∫0πf2(x) dx\displaystyle \int_{0}^{\pi } \left|f'(x)\right|^{2}\, dx \geqslant \int_{0}^{\pi } f^{2}(x)\, dx; f(0)=f(π)=0f(0) = f(\pi ) = 0.

Problem 4928

Prove the equality:

1π∫02π(f(x)−σn(x))2 dx=1n2∑ν=1n−1ν2(aν2+bν2)+∑ν=n∞(aν2+bν2), \frac{1}{\pi } \int _{0}^{2\pi } \left(f(x)-\sigma _{n}(x)\right)^{2}\, dx = \frac{1}{n^{2}} \sum _{\nu =1}^{n-1} \nu ^{2}(a_{\nu }^{2}+b_{\nu }^{2})+\sum _{\nu =n}^{\infty } (a_{\nu }^{2}+b_{\nu }^{2}),

where σn(x)\sigma_{n}(x) is the Fejér sum of the function f(x)f(x) (see 4910).

?
Problem 4929

Prove that for a function f(x)f(x) satisfying the Cauchy--Lipschitz condition

∣f(x2)−f(x1)∣<A⋅∣x2−x1∣α, \left|f(x_{2})-f(x_{1})\right| < A \cdot \left|x_{2}-x_{1}\right|^{\alpha },

the inequality holds:

∑ν=n2n(aν2+bν2)<kn2α, \sum _{\nu =n}^{2n} (a_{\nu }^{2}+b_{\nu }^{2}) < \frac{k}{n^{2\alpha }},

where kk is a corresponding constant.

?
Problem 4930

Prove S.N. Bernstein's theorem: the Fourier series of a function satisfying the Cauchy--Lipschitz condition with α>12\alpha > \dfrac {1}{2} converges absolutely.

?
Problem 4931

The parametric equation of a closed curve can be put in the form

x=∑n=1∞(ancos⁡nt+bnsin⁡nt);y=∑n=1∞(αncos⁡nt+βnsin⁡nt). x = \sum _{n=1}^{\infty } (a_{n}\cos nt+b_{n}\sin nt); \qquad y = \sum _{n=1}^{\infty } (\alpha _{n}\cos nt+\beta _{n}\sin nt) .

Using the formulas ds2=dx2+dy2ds^{2} = dx^{2}+dy^{2} and

A=12∫Sx dy−y dx, A = \frac{1}{2} \int _{S} x\, dy-y\, dx,

where AA is the area bounded by the closed curve SS, prove the inequality 4πA⩽L24\pi A \leqslant L^{2} (LL being the length of the curve SS), which expresses that among all closed curves of length LL the circle has the greatest area (Hurwitz's proof).

?
Problem 4932

Expand in Chebyshev polynomials Tn(x)T_{n}(x) the function defined on the interval (−1,1)(-1,1) by the condition f(x)=x3f(x) = x^{3}.

?
Remark.
?

The Chebyshev polynomials are defined by the formula Tn(x)=12n−1cos⁡(narccos⁡x)T_{n}(x) = \dfrac {1}{2^{n-1}}\cos (n\arccos x) and have the orthogonality property: for n≠mn \neq m, ∫−11Tn(x)Tm(x) dx1−x2=0\displaystyle \int_{-1}^{1} \frac{T_{n}(x)T_{m}(x)\, dx}{\sqrt{1-x^{2}}} = 0, while for n=mn = m the right-hand side equals π22n−1\dfrac {\pi }{2^{2n-1}}.

Problem 4933

Expand in Chebyshev polynomials Tn(x)T_{n}(x) the function defined on the interval (−1,1)(-1,1) by the conditions:

f(x)={0,−1<x<0;1,0<x<1. f(x) = \begin{cases} 0, & -1 < x < 0; \\ 1, & 0 < x < 1 . \end{cases}
?
Remark.
?

The Chebyshev polynomials are defined by the formula Tn(x)=12n−1cos⁡(narccos⁡x)T_{n}(x) = \dfrac {1}{2^{n-1}}\cos (n\arccos x) and have the orthogonality property: for n≠mn \neq m, ∫−11Tn(x)Tm(x) dx1−x2=0\displaystyle \int_{-1}^{1} \frac{T_{n}(x)T_{m}(x)\, dx}{\sqrt{1-x^{2}}} = 0, while for n=mn = m the right-hand side equals π22n−1\dfrac {\pi }{2^{2n-1}}.

Problem 4934

Expand in Chebyshev polynomials Tn(x)T_{n}(x) the function defined on the interval (−1,1)(-1,1) by the condition f(x)=∣x∣f(x) = \left|x\right|.

?
Remark.
?

The Chebyshev polynomials are defined by the formula Tn(x)=12n−1cos⁡(narccos⁡x)T_{n}(x) = \dfrac {1}{2^{n-1}}\cos (n\arccos x) and have the orthogonality property: for n≠mn \neq m, ∫−11Tn(x)Tm(x) dx1−x2=0\displaystyle \int_{-1}^{1} \frac{T_{n}(x)T_{m}(x)\, dx}{\sqrt{1-x^{2}}} = 0, while for n=mn = m the right-hand side equals π22n−1\dfrac {\pi }{2^{2n-1}}.

Problem 4935

Expand in Chebyshev polynomials Tn(x)T_{n}(x) the function defined on the interval (−1,1)(-1,1) by the condition f(x)=x2nf(x) = x^{2n}.

?
Remark.
?

The Chebyshev polynomials are defined by the formula Tn(x)=12n−1cos⁡(narccos⁡x)T_{n}(x) = \dfrac {1}{2^{n-1}}\cos (n\arccos x) and have the orthogonality property: for n≠mn \neq m, ∫−11Tn(x)Tm(x) dx1−x2=0\displaystyle \int_{-1}^{1} \frac{T_{n}(x)T_{m}(x)\, dx}{\sqrt{1-x^{2}}} = 0, while for n=mn = m the right-hand side equals π22n−1\dfrac {\pi }{2^{2n-1}}.

Problem 4936

Expand, for −1<x<1-1 < x < 1, in Legendre polynomials the function defined by the conditions:

f(x)={0,−1<x<0;1,0<x<1. f(x) = \begin{cases} 0, & -1 < x < 0; \\ 1, & 0 < x < 1 . \end{cases}
?
Remark.
?

The Legendre polynomials are defined by the formula Pn(x)=12n⋅n!dn(x2−1)ndxnP_{n}(x) = \dfrac {1}{2^{n} \cdot n!}\dfrac {d^{n}(x^{2}-1)^{n}}{dx^{n}} and have the orthogonality property: for n≠mn \neq m, ∫−11Pn(x)Pm(x) dx=0\displaystyle \int_{-1}^{1} P_{n}(x)P_{m}(x)\, dx = 0, while for n=mn = m the right-hand side equals 22n+1\dfrac {2}{2n+1}.

Problem 4937

Expand, for −1<x<1-1 < x < 1, in Legendre polynomials the function defined by the condition f(x)=∣x∣f(x) = \left|x\right|.

?
Remark.
?

The Legendre polynomials are defined by the formula Pn(x)=12n⋅n!dn(x2−1)ndxnP_{n}(x) = \dfrac {1}{2^{n} \cdot n!}\dfrac {d^{n}(x^{2}-1)^{n}}{dx^{n}} and have the orthogonality property: for n≠mn \neq m, ∫−11Pn(x)Pm(x) dx=0\displaystyle \int_{-1}^{1} P_{n}(x)P_{m}(x)\, dx = 0, while for n=mn = m the right-hand side equals 22n+1\dfrac {2}{2n+1}.

Problem 4938

Expand in Laguerre polynomials Ln(x)L_{n}(x), for x>0x > 0, the function f(x)=e−axf(x) = e^{-ax}.

?
Remark.
?

The Abel--Laguerre polynomials are defined by the formula Ln(x)=exn!dnxne−xdxnL_{n}(x) = \dfrac {e^{x}}{n!}\dfrac {d^{n}x^{n}e^{-x}}{dx^{n}} and have the orthogonality property: for n≠mn \neq m, ∫0∞e−xLn(x)Lm(x) dx=0\displaystyle \int_{0}^{\infty } e^{-x}L_{n}(x)L_{m}(x)\, dx = 0, while for n=mn = m the right-hand side equals 11.

Problem 4939

Expand in Laguerre polynomials Ln(x)L_{n}(x), for x>0x > 0, the function f(x)=xnf(x) = x^{n}.

?
Remark.
?

The Abel--Laguerre polynomials are defined by the formula Ln(x)=exn!dnxne−xdxnL_{n}(x) = \dfrac {e^{x}}{n!}\dfrac {d^{n}x^{n}e^{-x}}{dx^{n}} and have the orthogonality property: for n≠mn \neq m, ∫0∞e−xLn(x)Lm(x) dx=0\displaystyle \int_{0}^{\infty } e^{-x}L_{n}(x)L_{m}(x)\, dx = 0, while for n=mn = m the right-hand side equals 11.

Problem 4940

Expand in Laguerre polynomials Ln(x)L_{n}(x), for x>0x > 0, the function f(x)=sin⁡axf(x) = \sin ax.

?
Remark.
?

The Abel--Laguerre polynomials are defined by the formula Ln(x)=exn!dnxne−xdxnL_{n}(x) = \dfrac {e^{x}}{n!}\dfrac {d^{n}x^{n}e^{-x}}{dx^{n}} and have the orthogonality property: for n≠mn \neq m, ∫0∞e−xLn(x)Lm(x) dx=0\displaystyle \int_{0}^{\infty } e^{-x}L_{n}(x)L_{m}(x)\, dx = 0, while for n=mn = m the right-hand side equals 11.

Problem 4941

Expand in Laguerre polynomials Ln(x)L_{n}(x), for x>0x > 0, the function f(x)=cos⁡axf(x) = \cos ax.

?
Remark.
?

The Abel--Laguerre polynomials are defined by the formula Ln(x)=exn!dnxne−xdxnL_{n}(x) = \dfrac {e^{x}}{n!}\dfrac {d^{n}x^{n}e^{-x}}{dx^{n}} and have the orthogonality property: for n≠mn \neq m, ∫0∞e−xLn(x)Lm(x) dx=0\displaystyle \int_{0}^{\infty } e^{-x}L_{n}(x)L_{m}(x)\, dx = 0, while for n=mn = m the right-hand side equals 11.

Problem 4942

Expand in Chebyshev--Hermite polynomials, for −∞<x<∞-\infty < x < \infty, the function

sign⁡x={−1for x<0;1for x>0. \operatorname {sign} x = \begin{cases} -1 & \text{for } x < 0; \\ 1 & \text{for } x > 0 . \end{cases}
?
Remark.
?

The Chebyshev--Hermite polynomials are defined (in the present chapter) by the formula Hn(x)=ex2/2n!dne−x2/2dxnH_{n}(x) = \dfrac {e^{x^{2}/2}}{n!}\dfrac {d^{n}e^{-x^{2}/2}}{dx^{n}} and have the orthogonality property: for n≠mn \neq m, ∫−∞∞e−x2/2Hn(x)Hm(x) dx=0\displaystyle \int_{-\infty }^{\infty } e^{-x^{2}/2}H_{n}(x)H_{m}(x)\, dx = 0, while for n=mn = m the right-hand side equals 2πn!\dfrac {\sqrt{2\pi }}{n!}.

Problem 4943

Expand in Chebyshev--Hermite polynomials, for −∞<x<∞-\infty < x < \infty, the function f(x)=∣x∣f(x) = \left|x\right|.

?
Remark.
?

The Chebyshev--Hermite polynomials are defined (in the present chapter) by the formula Hn(x)=ex2/2n!dne−x2/2dxnH_{n}(x) = \dfrac {e^{x^{2}/2}}{n!}\dfrac {d^{n}e^{-x^{2}/2}}{dx^{n}} and have the orthogonality property: for n≠mn \neq m, ∫−∞∞e−x2/2Hn(x)Hm(x) dx=0\displaystyle \int_{-\infty }^{\infty } e^{-x^{2}/2}H_{n}(x)H_{m}(x)\, dx = 0, while for n=mn = m the right-hand side equals 2πn!\dfrac {\sqrt{2\pi }}{n!}.

Problem 4944

Expand in Chebyshev--Hermite polynomials, for −∞<x<∞-\infty < x < \infty, the function f(x)=e−axf(x) = e^{-ax}.

?
Remark.
?

The Chebyshev--Hermite polynomials are defined (in the present chapter) by the formula Hn(x)=ex2/2n!dne−x2/2dxnH_{n}(x) = \dfrac {e^{x^{2}/2}}{n!}\dfrac {d^{n}e^{-x^{2}/2}}{dx^{n}} and have the orthogonality property: for n≠mn \neq m, ∫−∞∞e−x2/2Hn(x)Hm(x) dx=0\displaystyle \int_{-\infty }^{\infty } e^{-x^{2}/2}H_{n}(x)H_{m}(x)\, dx = 0, while for n=mn = m the right-hand side equals 2πn!\dfrac {\sqrt{2\pi }}{n!}.

Problem 4945

Expand in Chebyshev--Hermite polynomials, for −∞<x<∞-\infty < x < \infty, the function f(x)=cos⁡axf(x) = \cos ax.

?
Remark.
?

The Chebyshev--Hermite polynomials are defined (in the present chapter) by the formula Hn(x)=ex2/2n!dne−x2/2dxnH_{n}(x) = \dfrac {e^{x^{2}/2}}{n!}\dfrac {d^{n}e^{-x^{2}/2}}{dx^{n}} and have the orthogonality property: for n≠mn \neq m, ∫−∞∞e−x2/2Hn(x)Hm(x) dx=0\displaystyle \int_{-\infty }^{\infty } e^{-x^{2}/2}H_{n}(x)H_{m}(x)\, dx = 0, while for n=mn = m the right-hand side equals 2πn!\dfrac {\sqrt{2\pi }}{n!}.

Problem 4946

Expand in Chebyshev--Hermite polynomials, for −∞<x<∞-\infty < x < \infty, the function f(x)=sin⁡axf(x) = \sin ax.

?
Remark.
?

The Chebyshev--Hermite polynomials are defined (in the present chapter) by the formula Hn(x)=ex2/2n!dne−x2/2dxnH_{n}(x) = \dfrac {e^{x^{2}/2}}{n!}\dfrac {d^{n}e^{-x^{2}/2}}{dx^{n}} and have the orthogonality property: for n≠mn \neq m, ∫−∞∞e−x2/2Hn(x)Hm(x) dx=0\displaystyle \int_{-\infty }^{\infty } e^{-x^{2}/2}H_{n}(x)H_{m}(x)\, dx = 0, while for n=mn = m the right-hand side equals 2πn!\dfrac {\sqrt{2\pi }}{n!}.

Problem 4947

Represent as a Fourier integral the function

f(x)={e−mx,x>0;emx,x<0;m>0. f(x) = \begin{cases} e^{-mx}, & x > 0; \\ e^{mx}, & x < 0; \quad m > 0 . \end{cases}
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Problem 4948

Represent as a Fourier integral the function

f(x)={e−mx,x>0;−emx,x<0;m>0. f(x) = \begin{cases} e^{-mx}, & x > 0; \\ -e^{mx}, & x < 0; \quad m > 0 . \end{cases}
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Problem 4949

Represent as a Fourier integral the function

f(x)={e−mx,x>0;0,x<0;m>0. f(x) = \begin{cases} e^{-mx}, & x > 0; \\ 0, & x < 0; \quad m > 0 . \end{cases}
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Problem 4950

Represent as a Fourier integral the function

f(x)={1,∣x∣<τ;0,∣x∣>τ. f(x) = \begin{cases} 1, & \left|x\right| < \tau ; \\ 0, & \left|x\right| > \tau . \end{cases}
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Problem 4951

Represent as a Fourier integral the function

f(x)={1,0<x<τ;−1,−τ<x<0;0,∣x∣>τ. f(x) = \begin{cases} 1, & 0 < x < \tau ; \\ -1, & -\tau < x < 0; \\ 0, & \left|x\right| > \tau . \end{cases}
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Problem 4952

Represent as a Fourier integral the function

f(x)={A(1−∣x−x0∣a),∣x−x0∣<a;0,∣x−x0∣>a. f(x) = \begin{cases} A\left(1-\dfrac {\left|x-x_{0}\right|}{a}\right), & \left|x-x_{0}\right| < a; \\ 0, & \left|x-x_{0}\right| > a . \end{cases}
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Problem 4953

Represent as a Fourier integral the function

f(x)={e−kxsin⁡ωx,x>0;0,x<0;k>0. f(x) = \begin{cases} e^{-kx}\sin \omega x, & x > 0; \\ 0, & x < 0; \quad k > 0 . \end{cases}
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Problem 4954

Represent as a Fourier integral the function

f(x)={sin⁡x,0<x<nπ;n an integer;0,x<0; x>nπ. f(x) = \begin{cases} \sin x, & 0 < x < n\pi ; \quad n \text{ an integer}; \\ 0, & x < 0; \, x > n\pi . \end{cases}
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