Section 11.3

Computing Definite Integrals by Integration and Substitution

[26/8%]
Show
LaTeX
Problem 4598

Prove the equalities:

∫0∞dx1+x3=2π33. \int _{0}^{\infty } \frac{dx}{1+x^{3}} = \frac{2\pi }{3\sqrt{3}} .
?
Problem 4599

Prove the equalities:

∫01x2+1x4+1 dx=π22. \int _{0}^{1} \frac{x^{2}+1}{x^{4}+1}\, dx = \frac{\pi }{2\sqrt{2}} .
?
Problem 4600

Prove the equalities:

∫−11dx(a−x)1−x2=πa2−1;a>1. \int _{-1}^{1} \frac{dx}{(a-x)\sqrt{1-x^{2}}} = \frac{\pi }{\sqrt{a^{2}-1}}; \quad a > 1 .
?
Problem 4601

Prove the equalities:

∫−11x4 dx(x2+1)1−x2=π2(2−1). \int _{-1}^{1} \frac{x^{4}\, dx}{(x^{2}+1)\sqrt{1-x^{2}}} = \frac{\pi }{2}(\sqrt{2}-1) .
?
Problem 4602

Prove the equalities:

∫abx−ab−x x dx=π8(b−a)(a+3b). \int _{a}^{b} \sqrt{\frac{x-a}{b-x}}\, x\, dx = \frac{\pi }{8}(b-a)(a+3b) .
?
Problem 4603

Prove the equalities:

∫0ax2a2−x2 dx=πa416. \int _{0}^{a} x^{2}\sqrt{a^{2}-x^{2}}\, dx = \frac{\pi a^{4}}{16} .
?
Problem 4604

Prove the equalities:

∫02πdxa+bcos⁡x=2∫0πdxa+bcos⁡x=2πa2−b2;∣b∣<a. \int _{0}^{2\pi } \frac{dx}{a+b\cos x} = 2\int _{0}^{\pi } \frac{dx}{a+b\cos x} = \frac{2\pi }{\sqrt{a^{2}-b^{2}}}; \quad \left|b\right| < a .
?
Problem 4605

Prove the equalities:

∫−11dxx2−2xcos⁡α+1=2π2∣sin⁡α∣. \int _{-1}^{1} \frac{dx}{x^{2}-2x\cos \alpha +1} = \frac{2\pi }{2\left|\sin \alpha \right|} .
?
Problem 4606

Prove the equalities:

∫0πsin⁡x dx1+2acos⁡x+a2={2for ∣a∣⩽1,2∣a∣for ∣a∣>1. \int _{0}^{\pi } \frac{\sin x\, dx}{\sqrt{1+2a\cos x+a^{2}}} = \begin{cases} 2 & \text{for } \left|a\right| \leqslant 1, \\ \dfrac {2}{\left|a\right|} & \text{for } \left|a\right| > 1 . \end{cases}
?
Problem 4607

Prove the equalities:

∫0πsin⁡2x dx1+2acos⁡x+a2={π2for ∣a∣<1,π2a2for ∣a∣>1. \int _{0}^{\pi } \frac{\sin ^{2} x\, dx}{1+2a\cos x+a^{2}} = \begin{cases} \dfrac {\pi }{2} & \text{for } \left|a\right| < 1, \\ \dfrac {\pi }{2a^{2}} & \text{for } \left|a\right| > 1 . \end{cases}
?
Problem 4608

Prove the equalities:

∫−11dx(1−2ax+a2)(1−2bx+b2)=1abln⁡1+ab1−ab; \int _{-1}^{1} \frac{dx}{\sqrt{(1-2ax+a^{2})(1-2bx+b^{2})}} = \frac{1}{\sqrt{ab}}\ln \frac{1+\sqrt{ab}}{1-\sqrt{ab}}; ∣a∣<1,∣b∣<1,ab>0. \left|a\right| < 1, \quad \left|b\right| < 1, \quad ab > 0 .
?
Problem 4609

Prove the equalities:

∫0∞e−axcos⁡bx dx=aa2+b2,a>0. \int _{0}^{\infty } e^{-ax}\cos bx\, dx = \frac{a}{a^{2}+b^{2}}, \quad a > 0 .
?
Problem 4610

Prove the equalities:

∫0∞e−axsin⁡bx dx=ba2+b2,a>0. \int _{0}^{\infty } e^{-ax}\sin bx\, dx = \frac{b}{a^{2}+b^{2}}, \quad a > 0 .
?
Problem 4611

Prove the equalities:

∫0π/2dxa2sin⁡2x+b2cos⁡2x=π2ab. \int _{0}^{\pi /2} \frac{dx}{a^{2}\sin ^{2} x+b^{2}\cos ^{2} x} = \frac{\pi }{2ab} .
?
Problem 4612

Prove the equalities:

∫0π/2dx(a2sin⁡2x+b2cos⁡2x)2=π4a2+b2a3b3. \int _{0}^{\pi /2} \frac{dx}{(a^{2}\sin ^{2} x+b^{2}\cos ^{2} x)^{2}} = \frac{\pi }{4}\frac{a^{2}+b^{2}}{a^{3}b^{3}} .
?
Problem 4613

Prove the equalities:

∫0π/2dxtg⁡x=∫0π/2tg⁡x dx=π2. \int _{0}^{\pi /2} \frac{dx}{\sqrt{\tg x}} = \int _{0}^{\pi /2} \sqrt{\tg x}\, dx = \frac{\pi }{\sqrt{2}} .
?
Problem 4614

Using Euler's formulas

cos⁡x=exi+e−xi2,sin⁡x=exi−e−xi2i, \cos x = \frac{e^{xi}+e^{-xi}}{2}, \qquad \sin x = \frac{e^{xi}-e^{-xi}}{2i},

prove the equalities (mm and nn are positive integers):

∫0πsin⁡2nx dx=π22n(2n)!(n!)2. \int _{0}^{\pi } \sin ^{2n} x\, dx = \frac{\pi }{2^{2n}}\frac{(2n)!}{(n!)^{2}} .
?
Problem 4615

Using Euler's formulas, prove the equalities (mm and nn are positive integers):

∫0πcos⁡(4n+1)xcos⁡x dx=π. \int _{0}^{\pi } \frac{\cos (4n+1)x}{\cos x}\, dx = \pi .
?
Problem 4616

Using Euler's formulas, prove the equalities (mm and nn are positive integers):

∫0πsin⁡mxsin⁡x dx={0for m=2n;πfor m=2n+1. \int _{0}^{\pi } \frac{\sin mx}{\sin x}\, dx = \begin{cases} 0 & \text{for } m=2n; \\ \pi & \text{for } m=2n+1 . \end{cases}
?
Problem 4617

Using Euler's formulas, prove the equalities (mm and nn are positive integers):

∫0πcos⁡nxcos⁡nx dx=π2n. \int _{0}^{\pi } \cos ^{n} x \cos nx\, dx = \frac{\pi }{2^{n}} .
?
Problem 4618

Using Euler's formulas, prove the equalities (mm and nn are positive integers):

∫0πcos⁡nxsin⁡nx dx=0. \int _{0}^{\pi } \cos ^{n} x \sin nx\, dx = 0 .
?
Problem 4619

Using Euler's formulas, prove the equalities (mm and nn are positive integers):

∫0πcos⁡mxcos⁡nx dx={0,if m−n is odd;π2mCmm−n2,if m−n is even; \int _{0}^{\pi } \cos ^{m} x \cos nx\, dx = \begin{cases} 0, & \text{if } m-n \text{ is odd}; \\ \dfrac {\pi }{2^{m}} C_{m}^{\frac{m-n}{2}}, & \text{if } m-n \text{ is even}; \end{cases} Cmk=m!k!(m−k)!;m⩾k. C_{m}^{k} = \frac{m!}{k!(m-k)!}; \quad m \geqslant k .
?
Problem 4620

Using Euler's formulas, prove the equalities (mm and nn are positive integers):

∫0∞e−axcos⁡(2m−1)xcos⁡x dx=(−1)m+1a+2∑n=1m−1(−1)m+n−1aa2+4n2;a>0. \int _{0}^{\infty } e^{-ax}\frac{\cos (2m-1)x}{\cos x}\, dx = \frac{(-1)^{m+1}}{a}+2\sum _{n=1}^{m-1} \frac{(-1)^{m+n-1}a}{a^{2}+4n^{2}}; \quad a > 0 .
?
Problem 4621

In the following problems one should apply the formula for repeated integration by parts, which can be written in the form:

∫uv(n) dx=uv(n−1)−u′v(n−2)+u′′v(n−3)+…+(−1)νu(ν)v(n−ν−1)+(−1)ν+1∫u(ν+1)v(n−ν−1) dx. \int u v^{(n)}\, dx = u v^{(n-1)}-u'v^{(n-2)}+u''v^{(n-3)}+\ldots +(-1)^{\nu } u^{(\nu )} v^{(n-\nu -1)}+(-1)^{\nu +1} \int u^{(\nu +1)} v^{(n-\nu -1)}\, dx .

Prove the equalities:

∫−11Pn(x)f(x) dx=0,∫0∞e−xLn(x)f(x) dx=0, \int _{-1}^{1} P_{n}(x) f(x)\, dx = 0, \qquad \int _{0}^{\infty } e^{-x} L_{n}(x) f(x)\, dx = 0, ∫−∞∞e−x2Hn(x)f(x) dx=0, \int _{-\infty }^{\infty } e^{-x^{2}} H_{n}(x) f(x)\, dx = 0,

where f(x)f(x) is an arbitrary polynomial of degree n−1n-1, and the Legendre polynomials Pn(x)P_{n}(x), the Laguerre polynomials Ln(x)L_{n}(x), and the Chebyshev--Hermite polynomials Hn(x)H_{n}(x) are defined by the equalities:

Pn(x)=12nn!dn(x2−1)ndxn;Ln(x)=exdnxne−xdxn; P_{n}(x) = \frac{1}{2^{n} n!} \frac{d^{n}(x^{2}-1)^{n}}{dx^{n}}; \quad L_{n}(x) = e^{x} \frac{d^{n} x^{n} e^{-x}}{dx^{n}}; Hn(x)=ex2dne−x2dxn. H_{n}(x) = e^{x^{2}} \frac{d^{n} e^{-x^{2}}}{dx^{n}} .
?
Problem 4622

Find, for integer n>0n > 0, the value of the Euler integral

Γ(n)=∫0∞xn−1e−x dx. \Gamma (n) = \int _{0}^{\infty } x^{n-1} e^{-x}\, dx .
?
Problem 4623

Find, for integers p>0p > 0 and q>0q > 0, the value of the Euler integral

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