Section 11.7

Euler Integrals

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Problem 4720

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

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

There is a formula

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

By a change of variables, integrals of the form

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

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

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

Prove the equalities, for positive integers nn and mm:

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

Prove the equalities, for positive integers nn and mm:

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

Prove the equalities, for positive integers nn and mm:

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

Prove the equalities, for positive integers nn and mm:

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

Prove the equalities, for positive integers nn and mm:

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

Find the value of the integrals:

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

Find the value of the integrals:

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

Find the value of the integrals:

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

Find the value of the integrals:

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

Find the value of the integrals:

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

Find the value of the integrals:

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

Find the value of the integrals:

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

Find the value of the integrals:

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

Find the value of the integrals:

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

Find the value of the integrals:

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

Find the value of the integrals:

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

Find the value of the integrals:

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

Find the value of the integrals:

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

Find the value of the integrals:

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

Find the value of the integrals:

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

Find the value of the integrals:

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

Find the value of the integrals:

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

Find the value of the integrals:

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

Find the value of the integrals:

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

Find the value of the integrals:

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

Find the value of the integrals:

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

Find the value of the integrals:

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

Find the value of the integrals:

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

Find the value of the integrals:

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

Find the value of the integrals:

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

Find the value of the integrals:

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

Find the value of the integrals:

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

Find the value of the integrals:

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

Find the value of the integrals:

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

Prove the equalities:

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

Prove the equalities:

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

Prove the equalities:

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

Prove the equalities:

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

Prove the equalities:

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

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

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Problem 4760

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

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