Chapter 7

Integrals Depending on a Parameter

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Section 7.1Proper Integrals Depending on a Parameter[33/94%]
Problem 3711

Prove that the integral

F(y)=∫01f(x,y) dx F(y)=\int _{0}^{1}f(x,y)\, dx

of the discontinuous function f(x,y)=sgn⁡(x−y)f(x,y)=\operatorname {sgn}(x-y) is a continuous function. Plot the graph of the function u=F(y)u=F(y).

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Problem 3712

Investigate the continuity of the function

F(y)=∫01yf(x)x2+y2 dx F(y)=\int _{0}^{1}\dfrac {yf(x)}{x^{2}+y^{2}}\, dx

where f(x)f(x) is a positive continuous function on the closed interval [0,1]\left[0,1\right].

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Problem 3713

Find:

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(a)
(b)

lim⁡a→0∫−11x2+a2 dx\lim \limits_{a \to 0}\displaystyle \int_{-1}^{1}\sqrt{x^{2}+a^{2}}\, dx;

(c)

lim⁡a→0∫02x2cos⁡ax dx\lim \limits_{a \to 0}\displaystyle \int_{0}^{2}x^{2}\cos ax\, dx;

(d)

lim⁡n→∞∫01dx1+(1+xn)n\lim \limits_{n \to \infty }\displaystyle \int_{0}^{1}\dfrac {dx}{1+\left(1+\dfrac {x}{n}\right)^{n}}.

Problem 3713.1

lim⁡a→0∫a1+adx1+x2+a2\lim \limits_{a \to 0}\displaystyle \int_{a}^{1+a}\dfrac {dx}{1+x^{2}+a^{2}};

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Problem 3713.1

lim⁡a→0∫a1+adx1+x2+a2\lim \limits_{a \to 0}\displaystyle \int_{a}^{1+a}\dfrac {dx}{1+x^{2}+a^{2}};

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Problem 3714

Let the function f(x)f(x) be continuous on the closed interval [a,A]\left[a,A\right]. Prove that

lim⁡h→+01h∫ax[f(t+h)−f(t)] dt=f(x)−f(a)(a<x<A). \lim _{h \to +0}\dfrac {1}{h}\int _{a}^{x}\left[f(t+h)-f(t)\right]\, dt=f(x)-f(a) \quad (a<x<A) .
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Problem 3714.1
[NO TASK]
Problem 3714.1
[NO TASK]
Problem 3715

In the expression below, can the limit operation be carried out under the integral sign

lim⁡y→0∫01xy2e−x2y2 dx ? \lim _{y \to 0}\int _{0}^{1}\dfrac {x}{y^{2}}e^{-\frac{x^{2}}{y^{2}}}\, dx \, ?
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Problem 3716

When y=0y=0, can Leibniz's rule be used to compute the derivative of the function

F(y)=∫01ln⁡x2+y2 dx F(y)=\int _{0}^{1}\ln \sqrt{x^{2}+y^{2}}\, dx

?

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Problem 3717

If

F(x)=∫xx2e−xy2 dy, F(x)=\int _{x}^{x^{2}}e^{-xy^{2}}\, dy ,

compute F′(x)F'(x).

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Problem 3718

Let:

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(a)

F(a)=∫sin⁡acos⁡aea1−x2 dxF(a)=\displaystyle \int_{\sin a}^{\cos a}e^{a\sqrt{1-x^{2}}}\, dx;

(b)

F(a)=∫a+αb+αsin⁡axx dxF(a)=\displaystyle \int_{a+\alpha }^{b+\alpha }\dfrac {\sin ax}{x}\, dx;

(c)

F(a)=∫0aln⁡(1+ax)x dxF(a)=\displaystyle \int_{0}^{a}\dfrac {\ln (1+ax)}{x}\, dx;

(d)

F(a)=∫0af(x+a,x−a) dxF(a)=\displaystyle \int_{0}^{a}f(x+a,x-a)\, dx;

(e)

F(a)=∫0a2dx∫x−ax+asin⁡(x2+y2−a2) dyF(a)=\displaystyle \int_{0}^{a^{2}}dx\int_{x-a}^{x+a}\sin (x^{2}+y^{2}-a^{2})\, dy,

find F′(a)F'(a).

Problem 3719

If

F(x)=∫0x(x+y)f(y) dy, F(x)=\int _{0}^{x}(x+y)f(y)\, dy ,

where f(x)f(x) is a differentiable function, find F′′(x)F''(x).

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Problem 3720

Let:

F(x)=∫abf(y)∣x−y∣ dy, F(x)=\int _{a}^{b}f(y)\left|x-y\right|\, dy ,

where a<ba<b and f(y)f(y) is a differentiable function, find F′′(x)F''(x).

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Problem 3721

Let:

F(x)=1h2∫0hdξ∫0hf(x+ξ+η) dη(h>0), F(x)=\dfrac {1}{h^{2}}\int _{0}^{h}d\xi \int _{0}^{h}f(x+\xi +\eta )\, d\eta \quad (h>0) ,

where f(x)f(x) is a continuous function, find F′′(x)F''(x).

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Problem 3722

Let:

F(x)=∫0xf(t)(x−t)n−1 dt, F(x)=\int _{0}^{x}f(t)(x-t)^{n-1}\, dt ,

find F(n)(x)F^{(n)}(x).

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Problem 3722.1
[NO TASK]
Problem 3722.1
[NO TASK]
Problem 3723

On the interval 1⩽x⩽31 \leqslant x \leqslant 3, approximate the function f(x)=x2f(x)=x^{2} by the linear function a+bxa+bx so that

∫13(a+bx−x2)2 dx=min⁡. \int _{1}^{3}(a+bx-x^{2})^{2}\, dx=\min .
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Problem 3724

Given the condition that the mean square deviation between the functions a+bxa+bx and 1+x2\sqrt{1+x^{2}} on the given interval [0,1]\left[0,1\right] is minimal, find the approximate formula

1+x2≈a+bx(0⩽x⩽1). \sqrt{1+x^{2}} \approx a+bx \quad (0 \leqslant x \leqslant 1) .
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Problem 3725

Find the complete elliptic integrals

E(k)=∫0π21−k2sin⁡2φ dφ E(k)=\int _{0}^{\frac{\pi }{2}}\sqrt{1-k^{2}\sin ^{2}\varphi }\, d\varphi

and

F(k)=∫0π2dφ1−k2sin⁡2φ(0<k<1) F(k)=\int _{0}^{\frac{\pi }{2}}\dfrac {d\varphi }{\sqrt{1-k^{2}\sin ^{2}\varphi }} \quad (0<k<1)

and express their derivatives in terms of the functions E(k)E(k) and F(k)F(k).

Prove that E(k)E(k) satisfies the differential equation

E′′(k)+1kE′(k)+E(k)1−k2=0. E''(k)+\dfrac {1}{k}E'(k)+\dfrac {E(k)}{1-k^{2}}=0 .
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Problem 3726

Prove that: the Bessel function of integer order nn

Jn(x)=1π∫0πcos⁡(nφ−xsin⁡φ) dφ J_{n}(x)=\dfrac {1}{\pi }\int _{0}^{\pi }\cos (n\varphi -x\sin \varphi )\, d\varphi

satisfies Bessel's equation

x2Jn′′(x)+xJn′(x)+(x2−n2)Jn(x)=0. x^{2}J''_{n}(x)+xJ'_{n}(x)+(x^{2}-n^{2})J_{n}(x)=0 .
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Problem 3727

Let:

I(a)=∫0aφ(x) dxa−x, I(a)=\int _{0}^{a}\dfrac {\varphi (x)\, dx}{\sqrt{a-x}} ,

where the function φ(x)\varphi (x) and its derivative φ′(x)\varphi '(x) are continuous on the closed interval 0⩽x⩽a0 \leqslant x \leqslant a.

Prove that when 0<a<α0<a<\alpha,

I′(a)=φ(0)a+∫0aφ’(x)a−x dx. I'(a)=\dfrac {\varphi (0)}{\sqrt{a}}+\int _{0}^{a}\dfrac {\varphi ’(x)}{\sqrt{a-x}}\, dx .
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Problem 3728

Given the function

u(x)=∫01K(x,y)v(y) dy, u(x)=\int _{0}^{1}K(x,y)v(y)\, dy ,

where

K(x,y)={x(1−y),if x⩽y;y(1−x),if x⩾y, K(x,y)= \begin{cases} x(1-y) , & \text{if } x \leqslant y; \\ y(1-x) , & \text{if } x \geqslant y, \end{cases}

and v(y)v(y) are both continuous. Prove that the given function satisfies the equation

u′′(x)=−v(x)(0⩽x⩽1). u''(x)=-v(x) \quad (0 \leqslant x \leqslant 1) .
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Problem 3729

Let:

F(x,y)=∫xyxy(x−yz)f(z) dz, F(x,y)=\int _{\frac{x}{y}}^{xy}(x-yz)f(z)\, dz ,

where f(z)f(z) is a differentiable function, find Fxy′′(x,y)F''_{xy}(x,y).

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Problem 3730

Let f(x)f(x) be a twice-differentiable function and F(x)F(x) a differentiable function. Prove that the function

u(x,t)=12[f(x−at)+f(x+at)]+12a∫x−atx+atF(z) dz u(x,t)=\dfrac {1}{2}\left[f(x-at)+f(x+at)\right]+\dfrac {1}{2a}\int _{x-at}^{x+at}F(z)\, dz

satisfies the wave equation for a vibrating string

∂2u∂t2=a2∂2u∂x2 \dfrac {\partial ^{2}u}{\partial t^{2}}=a^{2}\dfrac {\partial ^{2}u}{\partial x^{2}}

together with the initial conditions: u(x,0)=f(x)u(x,0)=f(x), ut′(x,0)=F(x)u'_{t}(x,0)=F(x).

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Problem 3731

Prove that: if the function f(x)f(x) is continuous on the closed interval [0,l]\left[0,l\right] and (x−ξ)2+y2+z2≠0(x-\xi )^{2}+y^{2}+z^{2} \neq 0 when 0⩽ξ⩽l0 \leqslant \xi \leqslant l, then the function

u(x,y,z)=∫0lf(ξ) dξ(x−ξ)2+y2+z2 u(x,y,z)=\int _{0}^{l}\dfrac {f(\xi )\, d\xi }{\sqrt{(x-\xi )^{2}+y^{2}+z^{2}}}

satisfies Laplace's equation

∂2u∂x2+∂2u∂y2+∂2u∂z2=0. \dfrac {\partial ^{2}u}{\partial x^{2}}+\dfrac {\partial ^{2}u}{\partial y^{2}}+\dfrac {\partial ^{2}u}{\partial z^{2}}=0 .
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Problem 3732

Using differentiation with respect to a parameter, evaluate the integral

∫0π2ln⁡(a2sin⁡2x+b2cos⁡2x) dx. \int _{0}^{\frac{\pi }{2}}\ln (a^{2}\sin ^{2}x+b^{2}\cos ^{2}x)\, dx .
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Problem 3733

Using differentiation with respect to a parameter, evaluate the integral

∫0πln⁡(1−2acos⁡x+a2) dx. \int _{0}^{\pi }\ln (1-2a\cos x+a^{2})\, dx .
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Problem 3734

Using differentiation with respect to a parameter, compute the integral

∫0π2arctg⁡(atg⁡x)tg⁡x dx. \int _{0}^{\frac{\pi }{2}}\dfrac {\operatorname {arctg}(a\operatorname {tg}x)}{\operatorname {tg}x}\, dx .
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Problem 3735

Using differentiation with respect to a parameter, compute the integral

∫0π2ln⁡1+acos⁡x1−acos⁡x⋅dxcos⁡x(∣a∣<1). \int _{0}^{\frac{\pi }{2}}\ln \dfrac {1+a\cos x}{1-a\cos x}\cdot \dfrac {dx}{\cos x} \quad (\left|a\right|<1) .
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Problem 3736

Using the formula

arctg⁡xx=∫01dy1+x2y2, \dfrac {\operatorname {arctg}x}{x}=\int _{0}^{1}\dfrac {dy}{1+x^{2}y^{2}} ,

compute the integral ∫01arctg⁡xx⋅dx1−x2\displaystyle \int_{0}^{1}\dfrac {\operatorname {arctg}x}{x}\cdot \dfrac {dx}{\sqrt{1-x^{2}}}.

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Problem 3737

Using integration under the integral sign, compute the integral

∫01xb−xaln⁡x dx(a>0, b>0). \int _{0}^{1}\dfrac {x^{b}-x^{a}}{\ln x}\, dx \quad (a>0,\ b>0) .
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Problem 3738

Compute the integrals:

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(a)

∫01sin⁡(ln⁡1x)xb−xaln⁡x dx\displaystyle \int_{0}^{1}\sin \left(\ln \dfrac {1}{x}\right)\dfrac {x^{b}-x^{a}}{\ln x}\, dx;

(b)

∫01cos⁡(ln⁡1x)xb−xaln⁡x dx(a>0, b>0)\displaystyle \int_{0}^{1}\cos \left(\ln \dfrac {1}{x}\right)\dfrac {x^{b}-x^{a}}{\ln x}\, dx \quad (a>0,\ b>0).

Problem 3739

Let F(k)F(k) and E(k)E(k) be complete elliptic integrals (see Problem 3725). Prove the formulas:

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(a)

∫0kF(k)k dk=E(k)−k12F(k)\displaystyle \int_{0}^{k}F(k)k\, dk=E(k)-k_{1}^{2}F(k);

(b)

∫0kE(k)k dk=13[(1+k2)E(k)−k12F(k)]\displaystyle \int_{0}^{k}E(k)k\, dk=\dfrac {1}{3}\left[(1+k^{2})E(k)-k_{1}^{2}F(k)\right],

where k12=1−k2k_{1}^{2}=1-k^{2}.

Problem 3740

Prove the formula

∫0xxJ0(x) dx=xJ1(x), \int _{0}^{x}xJ_{0}(x)\, dx=xJ_{1}(x) ,

where J0(x)J_{0}(x) and J1(x)J_{1}(x) are the Bessel functions of order 00 and 11 respectively (see Problem 3726).

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Section 7.2Improper Integrals with a Parameter. Uniform Convergence of Integrals[52/83%]
Problem 3741

Find the domain of convergence of the integral:

∫0+∞e−ax1+x2 dx. \int _{0}^{+\infty }\dfrac {e^{-ax}}{1+x^{2}}\, dx .
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Problem 3742

Find the domain of convergence of the integral:

∫π+∞xcos⁡xxp+xq dx. \int _{\pi }^{+\infty }\dfrac {x\cos x}{x^{p}+x^{q}}\, dx .
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Problem 3743

Find the domain of convergence of the integral:

∫0+∞sin⁡xqxp dx. \int _{0}^{+\infty }\dfrac {\sin x^{q}}{x^{p}}\, dx .
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Problem 3744

Find the domain of convergence of the integral:

∫02dx∣ln⁡x∣p. \int _{0}^{2}\dfrac {dx}{\left|\ln x\right|^{p}} .
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Problem 3745

Find the domain of convergence of the integral:

∫01cos⁡11−x1−x2n dx. \int _{0}^{1}\dfrac {\cos \dfrac {1}{1-x}}{\sqrt[n]{1-x^{2}}}\, dx .
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Problem 3746

Find the domain of convergence of the integral:

∫0+∞sin⁡xxp+sin⁡x dx(p>0). \int _{0}^{+\infty }\dfrac {\sin x}{x^{p}+\sin x}\, dx \quad (p>0) .
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Problem 3747

Using the method of comparison with series, study the convergence of the integral

∫0+∞cos⁡xx+a dx. \int _{0}^{+\infty }\dfrac {\cos x}{x+a}\, dx .
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Problem 3748

Using the method of comparison with series, study the convergence of the integral

∫0+∞x dx1+xnsin⁡2x(n>0) \int _{0}^{+\infty }\dfrac {x\, dx}{1+x^{n}\sin ^{2}x} \quad (n>0)
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Problem 3749

Using the method of comparison with series, study the convergence of the integral

∫π+∞dxxpsin⁡2x3 \int _{\pi }^{+\infty }\dfrac {dx}{x^{p}\sqrt[3]{\sin ^{2}x}}
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Problem 3750

Using the method of comparison with series, study the convergence of the integral

∫0+∞sin⁡(x+x2)xn dx \int _{0}^{+\infty }\dfrac {\sin (x+x^{2})}{x^{n}}\, dx
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Problem 3751

State in positive terms what it means for the integral

∫a+∞f(x,y) dx \int _{a}^{+\infty }f(x,y)\, dx

to converge non-uniformly on the given interval (y1,y2)(y_{1},y_{2})?

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Problem 3752

Prove that if 1) the integral

∫a+∞f(x) dx \int _{a}^{+\infty }f(x)\, dx

converges, and 2) the function φ(x,y)\varphi (x,y) is bounded and monotone in xx, then the integral

∫a+∞f(x)φ(x,y) dx \int _{a}^{+\infty }f(x)\varphi (x,y)\, dx

converges uniformly (on the corresponding domain).

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Problem 3753

Prove that the uniformly convergent integral

I=∫1+∞e−1y2(x−1y)2 dx(0<y<1) I=\int _{1}^{+\infty }e^{-\frac{1}{y^{2}}\left(x-\frac{1}{y}\right)^{2}}\, dx \quad (0<y<1)

cannot be dominated by a convergent integral of a majorant function independent of the parameter.

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Problem 3754

Prove that the integral

I=∫0+∞ae−ax dx I=\int _{0}^{+\infty }ae^{-ax}\, dx
  1. converges uniformly on any interval 0<a⩽α⩽b0<a \leqslant \alpha \leqslant b; 2) converges non-uniformly on the interval 0⩽α⩽b0 \leqslant \alpha \leqslant b.
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(a)
(b)
Problem 3755

Prove that the Dirichlet integral

I=∫0+∞sin⁡αxx dx I=\int _{0}^{+\infty }\dfrac {\sin \alpha x}{x}\, dx
  1. converges uniformly on every closed interval [a,b]\left[a,b\right] not containing the value α=0\alpha =0; 2) converges non-uniformly on every closed interval [a,b]\left[a,b\right] containing the value α=0\alpha =0.
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Problem 3755.1
(a)
(b)
Problem 3755.1
(a)
(b)
Problem 3755.2
[NO TASK]
Problem 3755.2
[NO TASK]
Problem 3755.3
[NO TASK]
Problem 3755.3
[NO TASK]
Problem 3756

Investigate the integral

∫0+∞e−αxsin⁡x dx(0<α0⩽α<+∞) \int _{0}^{+\infty }e^{-\alpha x}\sin x\, dx \quad (0<\alpha _{0} \leqslant \alpha <+\infty )

for uniform convergence on the indicated interval.

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Problem 3757

Investigate the integral

∫1+∞xαe−x dx(a⩽α⩽b) \int _{1}^{+\infty }x^{\alpha }e^{-x}\, dx \quad (a \leqslant \alpha \leqslant b)

for uniform convergence on the indicated interval.

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Problem 3758

Investigate the integral

∫−∞+∞cos⁡αx1+x2 dx(−∞<α<+∞) \int _{-\infty }^{+\infty }\dfrac {\cos \alpha x}{1+x^{2}}\, dx \quad (-\infty <\alpha <+\infty )

for uniform convergence on the indicated interval.

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Problem 3759

Investigate the integral

∫0+∞dx(x+α)2+1(0⩽α<+∞) \int _{0}^{+\infty }\dfrac {dx}{(x+\alpha )^{2}+1} \quad (0 \leqslant \alpha <+\infty )

for uniform convergence on the indicated interval.

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Problem 3760

Investigate the integral

∫0+∞sin⁡xxe−ax dx(0⩽a<+∞) \int _{0}^{+\infty }\dfrac {\sin x}{x}e^{-ax}\, dx \quad (0 \leqslant a<+\infty )

for uniform convergence on the indicated interval.

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Problem 3760.1
[NO TASK]
Problem 3760.1
[NO TASK]
Problem 3761

Investigate the integral

∫1+∞e−axcos⁡xxp dx(0⩽a<+∞), \int _{1}^{+\infty }e^{-ax}\dfrac {\cos x}{x^{p}}\, dx \quad (0 \leqslant a<+\infty ) ,

where p>0p>0 is a constant, for uniform convergence on the indicated interval.

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Problem 3762

Investigate the integral

∫0+∞α e−αx2 dx(0⩽α<+∞) \int _{0}^{+\infty }\sqrt{\alpha }\, e^{-\alpha x^{2}}\, dx \quad (0 \leqslant \alpha <+\infty )

for uniform convergence on the indicated interval.

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Problem 3763

Investigate the integral ∫−∞+∞e−(x−α)2 dx\displaystyle \int_{-\infty }^{+\infty }e^{-(x-\alpha )^{2}}\, dx for uniform convergence on the indicated interval:

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(a)

a⩽α⩽ba \leqslant \alpha \leqslant b;

(b)

−∞<α<+∞-\infty <\alpha <+\infty.

Problem 3764

Investigate the integral

∫0+∞e−x2(1+y2)sin⁡x dy(−∞<x<+∞) \int _{0}^{+\infty }e^{-x^{2}(1+y^{2})}\sin x\, dy \quad (-\infty <x<+\infty )

for uniform convergence on the indicated interval.

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Problem 3765

Investigate the integral

∫0+∞sin⁡(x2)1+xp dx(p⩾0) \int _{0}^{+\infty }\dfrac {\sin (x^{2})}{1+x^{p}}\, dx \quad (p \geqslant 0)

for uniform convergence on the indicated interval.

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Problem 3765.1
[NO TASK]
Problem 3765.1
[NO TASK]
Problem 3766

Investigate the integral ∫01xp−1ln⁡q1x dx\displaystyle \int_{0}^{1}x^{p-1}\ln^{q}\dfrac {1}{x}\, dx for uniform convergence on the indicated interval:

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(a)

p⩾p0>0p \geqslant p_{0}>0;

(b)

p>0p>0 (q>−1q>-1).

Problem 3767

Investigate the integral

∫01xn1−x2 dx(0⩽n<+∞) \int _{0}^{1}\dfrac {x^{n}}{\sqrt{1-x^{2}}}\, dx \quad (0 \leqslant n<+\infty )

for uniform convergence on the indicated interval.

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Problem 3768

Investigate the integral

∫01sin⁡1x⋅dxxn(0<n<2) \int _{0}^{1}\sin \dfrac {1}{x}\cdot \dfrac {dx}{x^{n}} \quad (0<n<2)

for uniform convergence on the indicated interval.

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Problem 3769

Investigate the integral

∫02xa dx(x−1)(x−2)23(∣a∣<12) \int _{0}^{2}\dfrac {x^{a}\, dx}{\sqrt[3]{(x-1)(x-2)^{2}}} \quad \left(\left|a\right|<\dfrac {1}{2}\right)

for uniform convergence on the indicated interval.

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Problem 3770

Investigate the integral

∫01sin⁡αx∣x−α∣ dx(0⩽α⩽1) \int _{0}^{1}\dfrac {\sin \alpha x}{\sqrt{\left|x-\alpha \right|}}\, dx \quad (0 \leqslant \alpha \leqslant 1)

for uniform convergence on the indicated interval.

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Problem 3771

If an integral converges uniformly in some neighborhood of a given value of the parameter, we say the integral converges uniformly at that value of the parameter. Prove that the integral

I=∫0+∞a dx1+a2x2 I=\int _{0}^{+\infty }\dfrac {a\, dx}{1+a^{2}x^{2}}

converges uniformly at every value a≠0a \neq 0, but converges non-uniformly at a=0a=0.

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Problem 3772

In the following expression

lim⁡a→+0∫0+∞ae−ax dx \lim _{a \to +0}\int _{0}^{+\infty }ae^{-ax}\, dx

is it valid to move the limit inside the integral sign?

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Problem 3773

The function f(x)f(x) is integrable on the interval (0,+∞)(0,+\infty ). Prove the formula

lim⁡a→+0∫0+∞e−axf(x) dx=∫0+∞f(x) dx. \lim _{a \to +0}\int _{0}^{+\infty }e^{-ax}f(x)\, dx=\int _{0}^{+\infty }f(x)\, dx .
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Problem 3773.1
[NO TASK]
Problem 3773.1
[NO TASK]
Problem 3774

If f(x)f(x) is absolutely integrable on the interval (0,+∞)(0,+\infty ), prove that

lim⁡n→∞∫0+∞f(x)sin⁡nx dx=0. \lim _{n \to \infty }\int _{0}^{+\infty }f(x)\sin nx\, dx=0 .
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Problem 3775

Prove that if (1) on every finite interval (a,b)(a,b), f(x,y)\unifconvf(x,y0)f(x,y) \unifconv f(x,y_{0}), and (2) ∣f(x,y)∣⩽F(x)\left|f(x,y)\right| \leqslant F(x), where ∫a+∞F(x) dx<+∞\displaystyle \int_{a}^{+\infty }F(x)\, dx<+\infty, then

lim⁡y→y0∫a+∞f(x,y) dx=∫a+∞lim⁡y→y0f(x,y) dx. \lim _{y \to y_{0}}\int _{a}^{+\infty }f(x,y)\, dx=\int _{a}^{+\infty }\lim _{y \to y_{0}}f(x,y)\, dx .
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Remark.
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Condition (1) means: as y→y0y \to y_{0}, f(x,y)f(x,y) tends to f(x,y0)f(x,y_{0}) uniformly in xx on any finite interval (a,b)(a,b).

Problem 3776

By interchanging the integral sign with the limit sign, compute the integral

∫0+∞e−x2 dx=∫0+∞lim⁡n→∞[(1+x2n)−n]dx. \int _{0}^{+\infty }e^{-x^{2}}\, dx=\int _{0}^{+\infty }\lim _{n \to \infty }\left[\left(1+\dfrac {x^{2}}{n}\right)^{-n}\right]dx .
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Problem 3776.1
[NO TASK]
Problem 3776.1
[NO TASK]
Problem 3776.2
[NO TASK]
Problem 3776.2
[NO TASK]
Problem 3777

Prove that the integral

F(a)=∫0+∞e−(x−a)2 dx F(a)=\int _{0}^{+\infty }e^{-(x-a)^{2}}\, dx

is a continuous function of the parameter aa.

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Problem 3777.1
[NO TASK]
Problem 3777.1
[NO TASK]
Problem 3778

Find the discontinuities of the function

F(a)=∫0+∞sin⁡(1−a2)xx dx F(a)=\int _{0}^{+\infty }\dfrac {\sin (1-a^{2})x}{x}\, dx

Sketch the graph of the function y=F(a)y=F(a).

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Problem 3779

Investigate the continuity of the function

F(a)=∫0+∞x dx2+xa F(a)=\int _{0}^{+\infty }\dfrac {x\, dx}{2+x^{a}}

on the indicated interval: a⩾2a \geqslant 2.

?
Problem 3780

Investigate the continuity of the function

F(a)=∫1+∞cos⁡xxa dx F(a)=\int _{1}^{+\infty }\dfrac {\cos x}{x^{a}}\, dx

on the indicated interval: a>0a>0.

?
Problem 3781

Investigate the continuity of the function

F(a)=∫0πsin⁡xxa(π−x)a dx F(a)=\int _{0}^{\pi }\dfrac {\sin x}{x^{a}(\pi -x)^{a}}\, dx

on the indicated interval: 0<a<20<a<2.

?
Problem 3782

Investigate the continuity of the function

F(a)=∫0+∞e−x∣sin⁡x∣a dx F(a)=\int _{0}^{+\infty }\dfrac {e^{-x}}{\left|\sin x\right|^{a}}\, dx

on the indicated interval: 0<a<10<a<1.

?
Problem 3783

Investigate the continuity of the function

F(a)=∫0+∞ae−xa2 dx F(a)=\int _{0}^{+\infty }ae^{-xa^{2}}\, dx

on the indicated interval: −∞<a<+∞-\infty <a<+\infty.

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Section 7.3Change of variable in improper integrals. Differentiation and integration under the integral sign for improper integrals[59/97%]
Problem 3784

Using the formula

∫01xn−1 dx=1n(n>0). \int _{0}^{1}x^{n-1}\, dx=\dfrac {1}{n} \quad (n>0) .

compute the integral

I=∫01xn−1ln⁡mx dx, I=\int _{0}^{1}x^{n-1}\ln ^{m}x\, dx ,

where mm is a natural number.

?
Problem 3785

Using the formula

∫0+∞dxx2+a=π2a(a>0), \int _{0}^{+\infty }\dfrac {dx}{x^{2}+a}=\dfrac {\pi }{2\sqrt{a}} \quad (a>0) ,

compute the integral

I=∫0+∞dx(x2+a)n+1, I=\int _{0}^{+\infty }\dfrac {dx}{(x^{2}+a)^{n+1}} ,

where nn is a natural number.

?
Problem 3786

Prove that the Dirichlet integral

I(a)=∫0+∞sin⁡axx dx I(a)=\int _{0}^{+\infty }\dfrac {\sin ax}{x}\, dx

has a derivative when a≠0a \neq 0, but that this derivative cannot be found by applying Leibniz's rule.

?
Problem 3787

Prove that the function

F(a)=∫0+∞cos⁡x1+(x+a)2 dx F(a)=\int _{0}^{+\infty }\dfrac {\cos x}{1+(x+a)^{2}}\, dx

is continuous and differentiable on the region −∞<a<+∞-\infty <a<+\infty.

?
Problem 3788

Starting from the identity

e−ax−e−bxx=∫abe−xy dy \dfrac {e^{-ax}-e^{-bx}}{x}=\int _{a}^{b}e^{-xy}\, dy

compute the integral

∫0+∞e−ax−e−bxx dx(a>0, b>0). \int _{0}^{+\infty }\dfrac {e^{-ax}-e^{-bx}}{x}\, dx \quad (a>0,\ b>0) .
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Problem 3789

Prove the Frullani formula

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

where f(x)f(x) is a continuous function and the integral ∫A+∞f(x)x dx\displaystyle \int_{A}^{+\infty }\dfrac {f(x)}{x}\, dx is meaningful for every A>0A>0.

?
Problem 3790

Using the Frullani formula (see problem 3789: ∫0+∞f(ax)−f(bx)x dx=f(0)ln⁡ba\displaystyle \int_{0}^{+\infty }\dfrac {f(ax)-f(bx)}{x}\, dx=f(0)\ln \dfrac {b}{a}, where f(x)f(x) is continuous and the integral ∫A+∞f(x)x dx\displaystyle \int_{A}^{+\infty }\dfrac {f(x)}{x}\, dx is meaningful for every A>0A>0), compute the integral:

∫0+∞cos⁡ax−cos⁡bxx dx(a>0, b>0). \int _{0}^{+\infty }\dfrac {\cos ax-\cos bx}{x}\, dx \quad (a>0,\ b>0) .
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Problem 3791

Using the Frullani formula (see problem 3789: ∫0+∞f(ax)−f(bx)x dx=f(0)ln⁡ba\displaystyle \int_{0}^{+\infty }\dfrac {f(ax)-f(bx)}{x}\, dx=f(0)\ln \dfrac {b}{a}, where f(x)f(x) is continuous and the integral ∫A+∞f(x)x dx\displaystyle \int_{A}^{+\infty }\dfrac {f(x)}{x}\, dx is meaningful for every A>0A>0), compute the integral:

∫0+∞sin⁡ax−sin⁡bxx dx(a>0, b>0). \int _{0}^{+\infty }\dfrac {\sin ax-\sin bx}{x}\, dx \quad (a>0,\ b>0) .
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Problem 3792

Compute the integral

∫0+∞arctg⁡ax−arctg⁡bxx dx(a>0, b>0). \int _{0}^{+\infty }\dfrac {\operatorname {arctg}ax-\operatorname {arctg}bx}{x}\, dx \quad (a>0,\ b>0) .
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Problem 3793

Use differentiation with respect to a parameter to compute the integral

∫0+∞e−αx2−e−βx2x dx(α>0, β>0). \int _{0}^{+\infty }\dfrac {e^{-\alpha x^{2}}-e^{-\beta x^{2}}}{x}\, dx \quad (\alpha >0,\ \beta >0) .
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Problem 3794

Compute the integral

∫0+∞(e−αx−e−βxx)2 dx(α>0, β>0). \int _{0}^{+\infty }\left(\dfrac {e^{-\alpha x}-e^{-\beta x}}{x}\right)^{2}\, dx \quad (\alpha >0,\ \beta >0) .
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Problem 3795

Compute the integral

∫0+∞e−ax−e−βxxsin⁡mx dx(a>0, β>0). \int _{0}^{+\infty }\dfrac {e^{-ax}-e^{-\beta x}}{x}\sin mx\, dx \quad (a>0,\ \beta >0) .
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Problem 3796

Compute the integral

∫0+∞e−ax−e−βxxcos⁡mx dx(a>0, β>0). \int _{0}^{+\infty }\dfrac {e^{-ax}-e^{-\beta x}}{x}\cos mx\, dx \quad (a>0,\ \beta >0) .
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Problem 3797

Compute the integral

∫01ln⁡(1−a2x2)x21−x2 dx(∣a∣⩽1). \int _{0}^{1}\dfrac {\ln (1-a^{2}x^{2})}{x^{2}\sqrt{1-x^{2}}}\, dx \quad (\left|a\right| \leqslant 1) .
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Problem 3798

Compute the integral

∫01ln⁡(1−a2x2)1−x2 dx(∣a∣⩽1). \int _{0}^{1}\dfrac {\ln (1-a^{2}x^{2})}{\sqrt{1-x^{2}}}\, dx \quad (\left|a\right| \leqslant 1) .
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Problem 3799

Compute the integral

∫1+∞arctg⁡axx2x2−1 dx. \int _{1}^{+\infty }\dfrac {\operatorname {arctg}ax}{x^{2}\sqrt{x^{2}-1}}\, dx .
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Problem 3800

Compute the integral

∫0+∞ln⁡(a2+x2)β2+x2 dx. \int _{0}^{+\infty }\dfrac {\ln (a^{2}+x^{2})}{\beta ^{2}+x^{2}}\, dx .
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Problem 3801

Compute the integral

∫0+∞arctg⁡ax⋅arctg⁡βxx2 dx. \int _{0}^{+\infty }\dfrac {\operatorname {arctg}ax \cdot \operatorname {arctg}\beta x}{x^{2}}\, dx .
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Problem 3802

Compute the integral

∫0+∞ln⁡(1+a2x2)ln⁡(1+β2x2)x4 dx. \int _{0}^{+\infty }\dfrac {\ln (1+a^{2}x^{2})\ln (1+\beta ^{2}x^{2})}{x^{4}}\, dx .
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Problem 3803

Starting from the formula

I2=∫0+∞e−x2 dx∫0+∞xe−x2y2 dy I^{2}=\int _{0}^{+\infty }e^{-x^{2}}\, dx\int _{0}^{+\infty }xe^{-x^{2}y^{2}}\, dy

compute the Euler--Poisson integral

I=∫0+∞e−x2 dx. I=\int _{0}^{+\infty }e^{-x^{2}}\, dx .
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Problem 3804

Using the Euler--Poisson integral ∫0+∞e−x2 dx=π2\displaystyle \int_{0}^{+\infty }e^{-x^{2}}\, dx=\dfrac {\sqrt{\pi }}{2}, find the integral

∫−∞+∞e−(ax2+2bx+c) dx(a>0, ac−b2⩾0). \int _{-\infty }^{+\infty }e^{-(ax^{2}+2bx+c)}\, dx \quad (a>0,\ ac-b^{2} \geqslant 0) .
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Problem 3805

Using the result of problem 3804, ∫−∞+∞e−(ax2+2bx+c) dx=πaeb2−aca\displaystyle \int_{-\infty }^{+\infty }e^{-(ax^{2}+2bx+c)}\, dx=\sqrt{\dfrac {\pi }{a}}e^{\frac{b^{2}-ac}{a}} (a>0a>0), find the integral

∫−∞+∞(a1x2+2b1x+c1)e−(ax2+2bx+c) dx(a>0, ac−b2⩾0). \int _{-\infty }^{+\infty }(a_{1}x^{2}+2b_{1}x+c_{1})e^{-(ax^{2}+2bx+c)}\, dx \quad (a>0,\ ac-b^{2} \geqslant 0) .
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Problem 3806

Using the Euler--Poisson integral, find the integral

∫−∞+∞e−ax2ch⁡bx dx(a>0). \int _{-\infty }^{+\infty }e^{-ax^{2}}\operatorname {ch}bx\, dx \quad (a>0) .
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Problem 3807

Using the Euler--Poisson integral, find the integral

∫0+∞e−(x2+a2x2) dx(a>0). \int _{0}^{+\infty }e^{-\left(x^{2}+\frac{a^{2}}{x^{2}}\right)}\, dx \quad (a>0) .
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Problem 3808

Compute the integral

∫0+∞e−αx2−e−βx2x2 dx(α>0, β>0). \int _{0}^{+\infty }\dfrac {e^{-\alpha x^{2}}-e^{-\beta x^{2}}}{x^{2}}\, dx \quad (\alpha >0,\ \beta >0) .
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Problem 3809

Compute the integral

∫0+∞e−ax2cos⁡bx dx(a>0). \int _{0}^{+\infty }e^{-ax^{2}}\cos bx\, dx \quad (a>0) .
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Problem 3810

Compute the integral

∫0+∞xe−ax2sin⁡bx dx(a>0). \int _{0}^{+\infty }xe^{-ax^{2}}\sin bx\, dx \quad (a>0) .
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Problem 3811

Compute the integral

∫0+∞x2ne−x2cos⁡2bx dx, \int _{0}^{+\infty }x^{2n}e^{-x^{2}}\cos 2bx\, dx ,

where nn is a natural number.

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Problem 3811.1
[NO TASK]
Problem 3811.1
[NO TASK]
Problem 3812

Starting from the integral

I(α)=∫0+∞e−αxsin⁡βxx dx I(\alpha )=\int _{0}^{+\infty }e^{-\alpha x}\dfrac {\sin \beta x}{x}\, dx

compute the Dirichlet integral

D(β)=∫0+∞sin⁡βxx dx. D(\beta )=\int _{0}^{+\infty }\dfrac {\sin \beta x}{x}\, dx .
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Problem 3812.1
[NO TASK]
Problem 3812.1
[NO TASK]
Problem 3813

Using the Dirichlet integral ∫0+∞sin⁡βxx dx=π2sgn⁡β\displaystyle \int_{0}^{+\infty }\dfrac {\sin \beta x}{x}\, dx=\dfrac {\pi }{2}\operatorname {sgn}\beta and the Frullani formula, find the integral

∫0+∞e−ax2−cos⁡βxx2 dx(a>0). \int _{0}^{+\infty }\dfrac {e^{-ax^{2}}-\cos \beta x}{x^{2}}\, dx \quad (a>0) .
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Problem 3814

Compute the integral

∫0+∞sin⁡axsin⁡βxx dx. \int _{0}^{+\infty }\dfrac {\sin ax\sin \beta x}{x}\, dx .
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Problem 3815

Compute the integral

∫0+∞sin⁡axcos⁡βxx dx. \int _{0}^{+\infty }\dfrac {\sin ax\cos \beta x}{x}\, dx .
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Problem 3816

Compute the integral

∫0+∞sin⁡3axx dx. \int _{0}^{+\infty }\dfrac {\sin ^{3}ax}{x}\, dx .
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Problem 3817

Compute the integral

∫0+∞(sin⁡axx)2 dx. \int _{0}^{+\infty }\left(\dfrac {\sin ax}{x}\right)^{2}\, dx .
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Problem 3818

Compute the integral

∫0+∞(sin⁡axx)3 dx. \int _{0}^{+\infty }\left(\dfrac {\sin ax}{x}\right)^{3}\, dx .
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Problem 3819

Compute the integral

∫0+∞sin⁡4xx2 dx. \int _{0}^{+\infty }\dfrac {\sin ^{4}x}{x^{2}}\, dx .
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Problem 3820

Compute the integral

∫0+∞sin⁡4ax−sin⁡4βxx dx. \int _{0}^{+\infty }\dfrac {\sin ^{4}ax-\sin ^{4}\beta x}{x}\, dx .
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Problem 3821

Compute the integral

∫0+∞sin⁡(x2)x dx. \int _{0}^{+\infty }\dfrac {\sin (x^{2})}{x}\, dx .
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Problem 3822

Compute the integral

∫0+∞e−kxsin⁡axsin⁡βxx2 dx(k⩾0, a>0, β>0). \int _{0}^{+\infty }e^{-kx}\dfrac {\sin ax\sin \beta x}{x^{2}}\, dx \quad (k \geqslant 0,\ a>0,\ \beta >0) .
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Problem 3823

For different values of xx, find the Dirichlet discontinuous multiplier

D(x)=2π∫0+∞sin⁡λcos⁡λx dλλ, D(x)=\dfrac {2}{\pi }\int _{0}^{+\infty }\sin \lambda \cos \lambda x\, \dfrac {d\lambda }{\lambda } ,

and sketch the graph of the function y=D(x)y=D(x).

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Problem 3824

Compute the integrals:

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(a)

V.P.∫−∞+∞sin⁡axx+b dx\text{V.P.}\displaystyle \int_{-\infty }^{+\infty }\dfrac {\sin ax}{x+b}\, dx;

(b)

V.P.∫−∞+∞cos⁡axx+b dx\text{V.P.}\displaystyle \int_{-\infty }^{+\infty }\dfrac {\cos ax}{x+b}\, dx.

Problem 3825

Using the formula

11+x2=∫0+∞e−y(1+x2) dy, \dfrac {1}{1+x^{2}}=\int _{0}^{+\infty }e^{-y(1+x^{2})}\, dy ,

compute the Laplace integral

L=∫0+∞cos⁡ax1+x2 dx. L=\int _{0}^{+\infty }\dfrac {\cos ax}{1+x^{2}}\, dx .
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Problem 3826

Compute the integral

L1=∫0+∞xsin⁡ax1+x2 dx. L_{1}=\int _{0}^{+\infty }\dfrac {x\sin ax}{1+x^{2}}\, dx .
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Problem 3827

Compute the integral

∫0+∞sin⁡2x1+x2 dx. \int _{0}^{+\infty }\dfrac {\sin ^{2}x}{1+x^{2}}\, dx.
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Problem 3828

Compute the integral

∫0+∞cos⁡ax(1+x2)2 dx. \int _{0}^{+\infty }\dfrac {\cos ax}{(1+x^{2})^{2}}\, dx.
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Problem 3829

Compute the integral

∫−∞+∞cos⁡axax2+2bx+c dx(a>0, ac−b2>0). \int _{-\infty }^{+\infty }\dfrac {\cos ax}{ax^{2}+2bx+c}\, dx \quad (a>0,\ ac-b^{2}>0).
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Problem 3830

Using the formula

1x=2π∫0+∞e−xy2 dy(x>0), \dfrac {1}{\sqrt{x}}=\dfrac {2}{\sqrt{\pi }}\int _{0}^{+\infty }e^{-xy^{2}}\, dy \quad (x>0),

compute the Fresnel integrals

∫0+∞sin⁡(x2) dx=12∫0+∞sin⁡xx dx \int _{0}^{+\infty }\sin (x^{2})\, dx=\dfrac {1}{2}\int _{0}^{+\infty }\dfrac {\sin x}{\sqrt{x}}\, dx

and

∫0+∞cos⁡(x2) dx=12∫0+∞cos⁡xx dx. \int _{0}^{+\infty }\cos (x^{2})\, dx=\dfrac {1}{2}\int _{0}^{+\infty }\dfrac {\cos x}{\sqrt{x}}\, dx.
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Problem 3831

Find the value of the integral

∫−∞+∞sin⁡(ax2+2bx+c) dx(a≠0). \int _{-\infty }^{+\infty }\sin (ax^{2}+2bx+c)\, dx \quad (a \neq 0).
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Problem 3832

Find the value of the integral

∫−∞+∞sin⁡x2⋅cos⁡2ax dx. \int _{-\infty }^{+\infty }\sin x^{2} \cdot \cos 2ax\, dx.
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Problem 3833

Find the value of the integral

∫−∞+∞cos⁡x2⋅cos⁡2ax dx. \int _{-\infty }^{+\infty }\cos x^{2} \cdot \cos 2ax\, dx.
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Problem 3834

Prove the formulas:

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(a)

∫0+∞cos⁡axa2−x2 dx=π2asin⁡aa(a⩾0)\displaystyle \int_{0}^{+\infty }\dfrac {\cos ax}{a^{2}-x^{2}}\, dx=\dfrac {\pi }{2a}\sin aa \quad (a \geqslant 0);

(b)

∫0+∞xsin⁡axa2−x2 dx=−π2cos⁡aa(a>0)\displaystyle \int_{0}^{+\infty }\dfrac {x\sin ax}{a^{2}-x^{2}}\, dx=-\dfrac {\pi }{2}\cos aa \quad (a>0),

where a≠0a \neq 0, and the integral is to be understood in the Cauchy principal-value sense.

Problem 3835

For the function f(t)f(t), find the Laplace transform

F(p)=∫0+∞e−ptf(t) dt(p>0). F(p)=\int _{0}^{+\infty }e^{-pt}f(t)\, dt \quad (p>0).

Let:

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(a)

f(t)=tnf(t)=t^{n} (nn a natural number);

(b)

f(t)=tf(t)=\sqrt{t};

(c)

f(t)=eatf(t)=e^{at};

(d)

f(t)=te−atf(t)=te^{-at};

(e)

f(t)=cos⁡tf(t)=\cos t;

(f)

f(t)=1−e−ttf(t)=\dfrac {1-e^{-t}}{t};

(g)

f(t)=sin⁡atf(t)=\sin a\sqrt{t}.

Problem 3836

Prove the formula (the Lipschitz integral)

∫0+∞e−atJ0(bt) dt=1a2+b2(a>0), \int _{0}^{+\infty }e^{-at}J_{0}(bt)\, dt=\dfrac {1}{\sqrt{a^{2}+b^{2}}} \quad (a>0),

where J0(x)=1π∫0πcos⁡(xsin⁡φ) dφJ_{0}(x)=\dfrac {1}{\pi }\displaystyle \int_{0}^{\pi }\cos (x\sin \varphi )\, d\varphi is the Bessel function of order 0 (see problem 3726: J0(x)=1π∫0πcos⁡(xsin⁡φ) dφJ_{0}(x)=\dfrac {1}{\pi }\displaystyle \int_{0}^{\pi }\cos (x\sin \varphi )\, d\varphi is a solution of the Bessel equation xy′′+y′+xy=0xy''+y'+xy=0).

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Problem 3837

Find the Weierstrass transform

F(x)=1π∫−∞+∞e−(x−y)2f(y) dy. F(x)=\dfrac {1}{\sqrt{\pi }}\int _{-\infty }^{+\infty }e^{-(x-y)^{2}}f(y)\, dy .

Let:

?
(a)

f(y)=1f(y)=1;

(b)

f(y)=y2f(y)=y^{2};

(c)

f(y)=e2ayf(y)=e^{2ay};

(d)

f(y)=cos⁡ayf(y)=\cos ay.

Problem 3838

The Chebyshev--Hermite polynomials are defined by the formula

Hn(x)=(−1)nex2dndxn(e−x2)(n=0,1,2,…) H_{n}(x)=(-1)^{n}e^{x^{2}}\dfrac {d^{n}}{dx^{n}}(e^{-x^{2}}) \quad (n=0,1,2,\ldots )

Prove that

∫−∞+∞Hm(x)Hn(x)e−x2 dx={0,if m≠n,2nn!π,if m=n. \int _{-\infty }^{+\infty }H_{m}(x)H_{n}(x)e^{-x^{2}}\, dx= \begin{cases} 0 , & \text{if } m \neq n , \\ 2^{n}n!\sqrt{\pi } , & \text{if } m=n . \end{cases}
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Problem 3839

Compute the integral, which is of significant importance in probability theory,

φ(x)=12πσ1σ2∫−∞+∞e−ξ22σ12e−(x−ξ)22σ22 dξ(σ1>0, σ2>0). \varphi (x)=\dfrac {1}{2\pi \sigma _{1}\sigma _{2}}\int _{-\infty }^{+\infty }e^{-\frac{\xi ^{2}}{2\sigma _{1}^{2}}}e^{-\frac{(x-\xi )^{2}}{2\sigma _{2}^{2}}}\, d\xi \quad (\sigma _{1}>0,\ \sigma _{2}>0) .
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Problem 3840

Let the function f(x)f(x) be continuous and absolutely integrable on the interval (−∞,+∞)(-\infty ,+\infty ). Prove that the integral

u(x,t)=12aπt∫−∞+∞f(ξ)e−(ξ−x)24a2t dξ u(x,t)=\dfrac {1}{2a\sqrt{\pi t}}\int _{-\infty }^{+\infty }f(\xi )e^{-\frac{(\xi -x)^{2}}{4a^{2}t}}\, d\xi

satisfies the heat conduction equation

∂u∂t=a2∂2u∂x2 \dfrac {\partial u}{\partial t}=a^{2}\dfrac {\partial ^{2}u}{\partial x^{2}}

and the initial value condition

lim⁡t→+0u(x,t)=f(x). \lim _{t \to +0}u(x,t)=f(x) .
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Section 7.4Euler Integrals[42/100%]
Problem 3841

Prove that the Γ\Gamma function Γ(x)\Gamma (x) is continuous on the domain x>0x>0, and has continuous derivatives of every order.

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Problem 3842

Prove that the BB function B(x,y)B(x,y) is continuous on the domain x>0x>0, y>0y>0, and has continuous derivatives of every order.

?
Problem 3843

Use the Euler integrals to compute the integral

∫01x−x2 dx. \int _{0}^{1}\sqrt{x-x^{2}}\, dx .
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Problem 3844

Use the Euler integrals to compute the integral

∫0ax2a2−x2 dx(a>0). \int _{0}^{a}x^{2}\sqrt{a^{2}-x^{2}}\, dx \quad (a>0) .
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Problem 3845

Use the Euler integrals to compute the integral

∫0+∞x4(1+x)2 dx. \int _{0}^{+\infty }\dfrac {\sqrt[4]{x}}{(1+x)^{2}}\, dx .
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Problem 3846

Use the Euler integrals to compute the integral

∫0+∞dx1+x3. \int _{0}^{+\infty }\dfrac {dx}{1+x^{3}} .
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Problem 3847

Use the Euler integrals to compute the integral

∫0+∞x2 dx1+x4. \int _{0}^{+\infty }\dfrac {x^{2}\, dx}{1+x^{4}} .
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Problem 3848

Use the Euler integrals to compute the integral

∫0π2sin⁡6x⋅cos⁡4x dx. \int _{0}^{\frac{\pi }{2}}\sin ^{6}x \cdot \cos ^{4}x\, dx .
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Problem 3849

Use the Euler integrals to compute the integral

∫01dx1−xnn(n>1). \int _{0}^{1}\dfrac {dx}{\sqrt[n]{1-x^{n}}} \quad (n>1) .
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Problem 3850

Use the Euler integrals to compute the integral

∫0+∞x2ne−x2 dx(n a positive integer). \int _{0}^{+\infty }x^{2n}e^{-x^{2}}\, dx \quad (n \text{ a positive integer}) .
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Problem 3851

Determine the domain of existence of the integral

∫0+∞xm−11+xn dx(n>0) \int _{0}^{+\infty }\dfrac {x^{m-1}}{1+x^{n}}\, dx \quad (n>0)

and express this integral using the Euler integrals.

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Problem 3852

Determine the domain of existence of the integral

∫0+∞xm−1(1+x)n dx \int _{0}^{+\infty }\dfrac {x^{m-1}}{(1+x)^{n}}\, dx

and express this integral using the Euler integrals.

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Problem 3853

Determine the domain of existence of the integral

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

and express this integral using Euler integrals.

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Problem 3854

Determine the domain of existence of the integral

∫ab(x−a)m(b−x)n(x+c)m+n+2 dx(0<a<b, c>0) \int _{a}^{b}\dfrac {(x-a)^{m}(b-x)^{n}}{(x+c)^{m+n+2}}\, dx \quad (0<a<b,\ c>0)

and express this integral using Euler integrals.

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Problem 3855

Determine the domain of existence of the integral

∫01dx1−xmn(m>0) \int _{0}^{1}\dfrac {dx}{\sqrt[n]{1-x^{m}}} \quad (m>0)

and express this integral using Euler integrals.

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Problem 3856

Determine the domain of existence of the integral

∫0π2sin⁡mxcos⁡nx dx \int _{0}^{\frac{\pi }{2}}\sin ^{m}x\cos ^{n}x\, dx

and express this integral using Euler integrals.

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Problem 3857

Determine the domain of existence of the integral

∫0π2tan⁡nx dx \int _{0}^{\frac{\pi }{2}}\tan ^{n}x\, dx

and express this integral using Euler integrals.

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Problem 3858

Determine the domain of existence of the integral

∫0πsin⁡n−1x(1+kcos⁡x)n dx(0<∣k∣<1) \int _{0}^{\pi }\dfrac {\sin ^{n-1}x}{(1+k\cos x)^{n}}\, dx \quad (0<\left|k\right|<1)

and express this integral using Euler integrals.

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Problem 3859

Determine the domain of existence of the integral

∫0+∞e−xn dx(n>0) \int _{0}^{+\infty }e^{-x^{n}}\, dx \quad (n>0)

and express this integral using Euler integrals.

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Problem 3860

Determine the domain of existence of the integral

∫0+∞xme−xn dx \int _{0}^{+\infty }x^{m}e^{-x^{n}}\, dx

and express this integral using Euler integrals.

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Problem 3861

Determine the domain of existence of the integral

∫01(ln⁡1x)p dx \int _{0}^{1}\left(\ln \dfrac {1}{x}\right)^{p}\, dx

and express this integral using Euler integrals.

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Problem 3862

Determine the domain of existence of the integral

∫0+∞xpe−axln⁡x dx(a>0) \int _{0}^{+\infty }x^{p}e^{-ax}\ln x\, dx \quad (a>0)

and express this integral using Euler integrals.

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Problem 3863

Determine the domain of existence of the integral

∫0+∞xp−1ln⁡x1+x dx(p>0) \int _{0}^{+\infty }\dfrac {x^{p-1}\ln x}{1+x}\, dx \quad (p>0)

and express this integral using Euler integrals.

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Problem 3864

Determine the domain of existence of the integral

∫0+∞xp−1ln⁡2x1+x dx \int _{0}^{+\infty }\dfrac {x^{p-1}\ln ^{2}x}{1+x}\, dx

and express this integral using Euler integrals.

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Problem 3864.1

Determine the domain of existence of the integral

∫0∞xln⁡x1+x3 dx \int _{0}^{\infty }\dfrac {x\ln x}{1+x^{3}}\, dx

and express this integral using Euler integrals.

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Problem 3864.1

Determine the domain of existence of the integral

∫0∞xln⁡x1+x3 dx \int _{0}^{\infty }\dfrac {x\ln x}{1+x^{3}}\, dx

and express this integral using Euler integrals.

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Problem 3864.2

Determine the domain of existence of the integral

∫0∞ln⁡2x1+x4 dx \int _{0}^{\infty }\dfrac {\ln ^{2}x}{1+x^{4}}\, dx

and express this integral using Euler integrals.

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Problem 3864.2

Determine the domain of existence of the integral

∫0∞ln⁡2x1+x4 dx \int _{0}^{\infty }\dfrac {\ln ^{2}x}{1+x^{4}}\, dx

and express this integral using Euler integrals.

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Problem 3865

Determine the domain of existence of the integral

∫0+∞xp−1−xq−1(1+x)ln⁡x dx \int _{0}^{+\infty }\dfrac {x^{p-1}-x^{q-1}}{(1+x)\ln x}\, dx

and express this integral using Euler integrals.

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Problem 3866

Determine the domain of existence of the integral

∫01xp−1−x−p1−x dx(0<p<1) \int _{0}^{1}\dfrac {x^{p-1}-x^{-p}}{1-x}\, dx \quad (0<p<1)

and express this integral using Euler integrals.

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Remark.
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This integral can be regarded as

lim⁡ε→+0[B(p,ε)−B(1−p,ε)]. \lim _{\varepsilon \to +0}\left[B(p,\varepsilon )-B(1-p,\varepsilon )\right] .
Problem 3867

Use Euler integrals to evaluate the integral

∫0+∞sh⁡αxsh⁡βx dx(0<α<β). \int _{0}^{+\infty }\dfrac {\operatorname {sh}\alpha x}{\operatorname {sh}\beta x}\, dx \quad (0<\alpha <\beta ) .
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Problem 3868

Use Euler integrals to evaluate the integral

∫01ln⁡Γ(x) dx. \int _{0}^{1}\ln \Gamma (x)\, dx .
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Problem 3869

Use Euler integrals to evaluate the integral

∫aa+1ln⁡Γ(x) dx(a>0). \int _{a}^{a+1}\ln \Gamma (x)\, dx \quad (a>0) .
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Problem 3870

Use the Euler integrals to evaluate the integral

∫01ln⁡Γ(x)sin⁡πx dx. \int _{0}^{1}\ln \Gamma (x)\sin \pi x\, dx.
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Problem 3871

Use the Euler integrals to evaluate the integral

∫01ln⁡Γ(x)cos⁡2nπx dx(n is a natural number). \int _{0}^{1}\ln \Gamma (x)\cos 2n\pi x\, dx \quad (n \text{ is a natural number}).
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Problem 3872

Prove the identity

∫01dx1−x4⋅∫01x2 dx1−x4=π4. \int _{0}^{1}\dfrac {dx}{\sqrt{1-x^{4}}} \cdot \int _{0}^{1}\dfrac {x^{2}\, dx}{\sqrt{1-x^{4}}}=\dfrac {\pi }{4}.
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Problem 3873

Prove the identity

∫0+∞e−x4 dx⋅∫0+∞x2e−x4 dx=π82. \int _{0}^{+\infty }e^{-x^{4}}\, dx \cdot \int _{0}^{+\infty }x^{2}e^{-x^{4}}\, dx=\dfrac {\pi }{8\sqrt{2}}.
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Problem 3874

Prove the identity

∏m=1n∫0+∞xm−1e−xn dx=(1n)n+12(2π)n−12. \prod _{m=1}^{n}\int _{0}^{+\infty }x^{m-1}e^{-x^{n}}\, dx=\left(\dfrac {1}{n}\right)^{n+\frac{1}{2}}(2\pi )^{\frac{n-1}{2}}.
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Problem 3875

Prove the identity

lim⁡n→∞∫0+∞e−xn dx=1. \lim _{n \to \infty }\int _{0}^{+\infty }e^{-x^{n}}\, dx=1.
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Problem 3876

Using the identity 1xm=1Γ(m)∫0+∞tm−1e−xt dt\dfrac {1}{x^{m}}=\dfrac {1}{\Gamma (m)}\displaystyle \int_{0}^{+\infty }t^{m-1}e^{-xt}\, dt (x>0x>0), compute the integral

∫0+∞cos⁡axxm dx(0<m<1). \int _{0}^{+\infty }\dfrac {\cos ax}{x^{m}}\, dx \quad (0<m<1).
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Problem 3877

Using the identity 1xm=1Γ(m)∫0+∞tm−1e−xt dt\dfrac {1}{x^{m}}=\dfrac {1}{\Gamma (m)}\displaystyle \int_{0}^{+\infty }t^{m-1}e^{-xt}\, dt (x>0x>0), compute the integral

∫0+∞sin⁡axxm dx(0<m<2). \int _{0}^{+\infty }\dfrac {\sin ax}{x^{m}}\, dx \quad (0<m<2).
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Problem 3878

Prove the Euler formulas:

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(a)

∫0+∞tx−1e−λtcos⁡αcos⁡(λtsin⁡α) dt=Γ(x)λxcos⁡αx\displaystyle \int_{0}^{+\infty }t^{x-1}e^{-\lambda t\cos \alpha }\cos (\lambda t\sin \alpha )\, dt=\dfrac {\Gamma (x)}{\lambda^{x}}\cos \alpha x;

(b)

∫0+∞tx−1e−λtcos⁡αsin⁡(λtsin⁡α) dt=Γ(x)λxsin⁡αx\displaystyle \int_{0}^{+\infty }t^{x-1}e^{-\lambda t\cos \alpha }\sin (\lambda t\sin \alpha )\, dt=\dfrac {\Gamma (x)}{\lambda^{x}}\sin \alpha x,

where λ>0\lambda >0, x>0x>0, −π2<α<π2-\dfrac {\pi }{2}<\alpha <\dfrac {\pi }{2}.

Problem 3879

Find the arc length of the curve rn=ancos⁡nφr^{n}=a^{n}\cos n\varphi (a>0a>0, nn a natural number).

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Problem 3880

Find the area enclosed by the curve ∣x∣n+∣y∣n=an\left|x\right|^{n}+\left|y\right|^{n}=a^{n} (n>0n>0, a>0a>0).

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Section 7.5Fourier Integral Formulas[20/100%]
Problem 3881

Represent the function by its Fourier integral:

f(x)={1,if ∣x∣<1;0,if ∣x∣>1. f(x)= \begin{cases} 1, & \text{if } \left|x\right|<1; \\ 0, & \text{if } \left|x\right|>1. \end{cases}
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Problem 3882

Represent the function by its Fourier integral:

f(x)={sgn⁡x,if ∣x∣<1;0,if ∣x∣>1. f(x)= \begin{cases} \operatorname {sgn}x, & \text{if } \left|x\right|<1; \\ 0, & \text{if } \left|x\right|>1. \end{cases}
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Problem 3883

Represent the function by its Fourier integral:

f(x)=sgn⁡(x−a)−sgn⁡(x−b)(b>a). f(x)=\operatorname {sgn}(x-a)-\operatorname {sgn}(x-b) \quad (b>a).
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Problem 3884

Represent the function by its Fourier integral:

f(x)={h(1−∣x∣a),if ∣x∣⩽a;0,if ∣x∣>a. f(x)= \begin{cases} h\left(1-\dfrac {\left|x\right|}{a}\right), & \text{if } \left|x\right| \leqslant a; \\ 0, & \text{if } \left|x\right|>a. \end{cases}
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Problem 3885

Represent the function by its Fourier integral:

f(x)=1a2+x2(a>0). f(x)=\dfrac {1}{a^{2}+x^{2}} \quad (a>0).
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Problem 3886

Represent the function by its Fourier integral:

f(x)=xa2+x2(a>0). f(x)=\dfrac {x}{a^{2}+x^{2}} \quad (a>0).
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Problem 3887

Represent the function by its Fourier integral:

f(x)={sin⁡x,if ∣x∣⩽π;0,if ∣x∣>π. f(x)= \begin{cases} \sin x, & \text{if } \left|x\right| \leqslant \pi ; \\ 0, & \text{if } \left|x\right|>\pi . \end{cases}
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Problem 3888

Represent by a Fourier integral the function

f(x)={cos⁡x,if ∣x∣⩽π2;0,if ∣x∣>π2. f(x)= \begin{cases} \cos x , & \text{if } \left|x\right| \leqslant \dfrac {\pi }{2}; \\ 0 , & \text{if } \left|x\right|>\dfrac {\pi }{2}. \end{cases}
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Problem 3889

Represent by a Fourier integral the function

f(t)={Asin⁡ωt,if ∣t∣⩽2πnω;0,if ∣t∣>2πnω(n a natural number). f(t)= \begin{cases} A\sin \omega t , & \text{if } \left|t\right| \leqslant \dfrac {2\pi n}{\omega }; \\ 0 , & \text{if } \left|t\right|>\dfrac {2\pi n}{\omega } \end{cases} \quad (n \text{ a natural number}) .
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Problem 3890

Represent by a Fourier integral the function

f(x)=e−α∣x∣(α>0). f(x)=e^{-\alpha \left|x\right|} \quad (\alpha >0) .
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Problem 3891

Represent by a Fourier integral the function

f(x)=e−α∣x∣cos⁡βx(α>0). f(x)=e^{-\alpha \left|x\right|}\cos \beta x \quad (\alpha >0) .
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Problem 3892

Represent by a Fourier integral the function

f(x)=e−α∣x∣sin⁡βx(α>0). f(x)=e^{-\alpha \left|x\right|}\sin \beta x \quad (\alpha >0) .
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Problem 3893

Represent by a Fourier integral the function

f(x)=e−x2. f(x)=e^{-x^{2}} .
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Problem 3894

Represent by a Fourier integral the function

f(x)=xe−x2. f(x)=xe^{-x^{2}} .
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Problem 3895

Represent by a Fourier integral the function

f(x)=e−x(0<x<+∞). f(x)=e^{-x} \quad (0<x<+\infty ) .
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(a)

Extend it as an even function;

(b)

Extend it as an odd function.

Problem 3896

For the function f(x)=e−α∣x∣f(x)=e^{-\alpha \left|x\right|} (α>0\alpha >0), find its Fourier transform

F(x)=12π∫−∞+∞f(t)e−itx dt=12πlim⁡l→+∞∫−llf(t)e−itx dt. F(x)=\dfrac {1}{\sqrt{2\pi }}\int _{-\infty }^{+\infty }f(t)e^{-itx}\, dt=\dfrac {1}{\sqrt{2\pi }}\lim _{l \to +\infty }\int _{-l}^{l}f(t)e^{-itx}\, dt .
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Problem 3897

For the function f(x)=xe−α∣x∣f(x)=xe^{-\alpha \left|x\right|} (α>0\alpha >0), find its Fourier transform

F(x)=12π∫−∞+∞f(t)e−itx dt=12πlim⁡l→+∞∫−llf(t)e−itx dt. F(x)=\dfrac {1}{\sqrt{2\pi }}\int _{-\infty }^{+\infty }f(t)e^{-itx}\, dt=\dfrac {1}{\sqrt{2\pi }}\lim _{l \to +\infty }\int _{-l}^{l}f(t)e^{-itx}\, dt .
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Problem 3898

For the function f(x)=e−x22f(x)=e^{-\frac{x^{2}}{2}}, find its Fourier transform

F(x)=12π∫−∞+∞f(t)e−itx dt=12πlim⁡l→+∞∫−llf(t)e−itx dt. F(x)=\dfrac {1}{\sqrt{2\pi }}\int _{-\infty }^{+\infty }f(t)e^{-itx}\, dt=\dfrac {1}{\sqrt{2\pi }}\lim _{l \to +\infty }\int _{-l}^{l}f(t)e^{-itx}\, dt .
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Problem 3899

For the function f(x)=e−x22cos⁡αxf(x)=e^{-\frac{x^{2}}{2}}\cos \alpha x, find its Fourier transform

F(x)=12π∫−∞+∞f(t)e−itx dt=12πlim⁡l→+∞∫−llf(t)e−itx dt. F(x)=\dfrac {1}{\sqrt{2\pi }}\int _{-\infty }^{+\infty }f(t)e^{-itx}\, dt=\dfrac {1}{\sqrt{2\pi }}\lim _{l \to +\infty }\int _{-l}^{l}f(t)e^{-itx}\, dt .
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Problem 3900

Find the functions φ(x)\varphi (x) and ψ(x)\psi (x), if

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(a)

∫0+∞φ(y)cos⁡xy dy=11+x2\displaystyle \int_{0}^{+\infty }\varphi (y)\cos xy\, dy=\dfrac {1}{1+x^{2}};

(b)

∫0+∞ψ(y)sin⁡xy dy=e−x\displaystyle \int_{0}^{+\infty }\psi (y)\sin xy\, dy=e^{-x} (x>0x>0).