Section 11.8

Miscellaneous Problems

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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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