Section 11.4

Finding Integrals by Means of Reduction Formulas

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Problem 4624

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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Problem 4632

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

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

Find the integrals (mm and nn positive integers).

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

Find the integrals (mm and nn positive integers).

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

Find the integrals (mm and nn positive integers).

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

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

Find the integral:

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

Find the integral:

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

Find the integrals (mm and nn positive integers).

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

Find the integrals (mm and nn positive integers).

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

Find the integrals (mm and nn positive integers).

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

Find the integrals (mm and nn positive integers).

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

Prove the equality:

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

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

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Problem 4643

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

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