Section 7.3

Change of variable in improper integrals. Differentiation and integration under the integral sign for improper integrals

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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.

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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.

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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.

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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.

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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.

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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 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 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:

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(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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