Section 5.11

Integrals Along the Real Axis

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Problem 5.11.1

Let the complex valued functions fnf_{n}, n∈Zn \in \mathbb {Z}, be defined on R\mathbb {R} by

fn(x)=(x−i)nπ(x+i)n+1. f_{n}(x) = \frac{(x-i)^{n}}{\sqrt{\pi }(x+i)^{n+1}} .

Prove that these functions are orthonormal; that is,

∫−∞∞fm(x)fn(x)‾ dx={1if m=n0if m≠n. \int _{-\infty }^{\infty } f_{m}(x) \overline{f_{n}(x)} \, dx = \begin{cases} 1 & \text{if } m=n \\ 0 & \text{if } m \neq n. \end{cases}
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Problem 5.11.2

Evaluate the integral

∫0∞1−cos⁡axx2 dx \int _{0}^{\infty } \frac{1-\cos ax}{x^{2}} \, dx

for a∈Ra \in \mathbb {R}.

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Problem 5.11.3

Evaluate the integrals

I(t)=∫−∞∞eitx(x+i)2 dx,−∞<t<∞. I(t) = \int _{-\infty }^{\infty } \frac{e^{itx}}{(x+i)^{2}} \, dx , \qquad -\infty < t < \infty .
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Problem 5.11.4

Evaluate the integrals

F(t)=∫−∞∞e−itx(x+i)3 dx,−∞<t<∞. F(t) = \int _{-\infty }^{\infty } \frac{e^{-itx}}{(x+i)^{3}} \, dx , \qquad -\infty < t < \infty .
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Problem 5.11.5

Evaluate

∫−∞∞sin⁡2xx2 dx. \int _{-\infty }^{\infty } \frac{\sin ^{2}x}{x^{2}} \, dx .
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Problem 5.11.6

Evaluate

∫−∞∞sin⁡3xx3 dx. \int _{-\infty }^{\infty } \frac{\sin ^{3}x}{x^{3}} \, dx .
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Problem 5.11.7

Evaluate

∫−∞∞x3sin⁡x(1+x2)2 dx. \int _{-\infty }^{\infty } \frac{x^{3}\sin x}{(1+x^{2})^{2}} \, dx .
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Problem 5.11.8

Evaluate

∫−∞∞xsin⁡x(1+x2)2 dx. \int _{-\infty }^{\infty } \frac{x\sin x}{(1+x^{2})^{2}} \, dx .
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Problem 5.11.9

Let aa be a positive real number. Evaluate the improper integral

∫0∞sin⁡xx(x2+a2) dx. \int _{0}^{\infty } \frac{\sin x}{x(x^{2}+a^{2})} \, dx .
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Problem 5.11.10

Prove that

lim⁡R→∞∫−RRsin⁡xx−3i dx \lim _{R \to \infty } \int _{-R}^{R} \frac{\sin x}{x-3i} \, dx

exists and find its value.

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Problem 5.11.11

Evaluate

∫−∞∞sin⁡xx(x−π) dx. \int _{-\infty }^{\infty } \frac{\sin x}{x(x-\pi )} \, dx .
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Problem 5.11.12

Evaluate

∫−∞∞cos⁡x(1+x2)3 dx. \int _{-\infty }^{\infty } \frac{\cos x}{(1+x^{2})^{3}} \, dx .
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Problem 5.11.13

Evaluate the integral

∫−∞∞cos⁡kx1+x+x2 dx \int _{-\infty }^{\infty } \frac{\cos kx}{1+x+x^{2}} \, dx

where k≥0k \geq 0.

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Problem 5.11.14

Evaluate

∫−∞∞cos⁡πx4x2−1 dx. \int _{-\infty }^{\infty } \frac{\cos \pi x}{4x^{2}-1} \, dx .
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Problem 5.11.15

Evaluate

∫−∞∞cos⁡nxx4+1 dx. \int _{-\infty }^{\infty } \frac{\cos nx}{x^{4}+1} \, dx .
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Problem 5.11.16

Evaluate

∫0∞x2+1x4+1 dx. \int _{0}^{\infty } \frac{x^{2}+1}{x^{4}+1} \, dx .
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Problem 5.11.17

Evaluate the integrals

I(a)=∫−∞∞cos⁡3xa2+x2 dx I(a) = \int _{-\infty }^{\infty } \frac{\cos ^{3}x}{a^{2}+x^{2}} \, dx

for a>0a > 0.

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Problem 5.11.18

Evaluate

∫−∞∞xsin⁡xx2+4x+20 dx. \int _{-\infty }^{\infty } \frac{x\sin x}{x^{2}+4x+20} \, dx .
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Problem 5.11.19

Evaluate

∫−∞∞x−sin⁡xx3 dx. \int _{-\infty }^{\infty } \frac{x-\sin x}{x^{3}} \, dx .
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Problem 5.11.20

Evaluate

∫−∞∞dx(1+x+x2)2. \int _{-\infty }^{\infty } \frac{dx}{(1+x+x^{2})^{2}} .
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Problem 5.11.21

Prove that

∫0∞xα−11+x dx=πsin⁡πα. \int _{0}^{\infty } \frac{x^{\alpha -1}}{1+x} \, dx = \frac{\pi }{\sin \pi \alpha } .

What restrictions must be placed on α\alpha?

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Problem 5.11.22

Evaluate the integral

I=∫0∞dx1+x2. I = \int _{0}^{\infty } \frac{dx}{1+x^{2}} .
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Problem 5.11.23

Evaluate

∫0∞dx1+x5. \int _{0}^{\infty } \frac{dx}{1+x^{5}} .
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Problem 5.11.24

Evaluate

∫−∞∞dx1+x2n \int _{-\infty }^{\infty } \frac{dx}{1+x^{2n}}

where nn is a positive integer.

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Problem 5.11.25

Evaluate ∫−∞∞x2xn+1 dx\displaystyle \int_{-\infty }^{\infty } \frac{x^{2}}{x^{n}+1} \, dx, where n≥4n \geq 4 is an even integer.

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Problem 5.11.26

Prove that

∫0∞xex−e−x dx=π28. \int _{0}^{\infty } \frac{x}{e^{x}-e^{-x}} \, dx = \frac{\pi ^{2}}{8} .
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Problem 5.11.27

Evaluate

∫−∞∞e−ixx2−2x+4 dx. \int _{-\infty }^{\infty } \frac{e^{-ix}}{x^{2}-2x+4} \, dx .
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Problem 5.11.28

Evaluate

∫0∞log⁡x(x2+1)(x2+4) dx. \int _{0}^{\infty } \frac{\log x}{(x^{2}+1)(x^{2}+4)} \, dx .
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Problem 5.11.29

Evaluate

∫0∞log⁡xx2+1 dx. \int _{0}^{\infty } \frac{\log x}{x^{2}+1} \, dx .
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Problem 5.11.30

Evaluate

∫−∞∞(sech⁡x)2cos⁡λx dx \int _{-\infty }^{\infty } (\operatorname {sech} x)^{2} \cos \lambda x \, dx

where λ\lambda is a real constant and

sech⁡x=2ex+e−x. \operatorname {sech} x = \frac{2}{e^{x}+e^{-x}} .
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Problem 5.11.31

Prove that

∫0∞e−x2cos⁡(2bx) dx=π2e−b2. \int _{0}^{\infty } e^{-x^{2}} \cos (2bx) \, dx = \frac{\sqrt{\pi }}{2} e^{-b^{2}} .

What restrictions, if any, need be placed on bb?

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Problem 5.11.32

Evaluate

∫0∞e−x2cos⁡x2 dx. \int _{0}^{\infty } e^{-x^{2}} \cos x^{2} \, dx .
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Problem 5.11.33

Prove that

∫−∞∞e−(t−iγ)2/22π dt \int _{-\infty }^{\infty } \frac{e^{-(t-i\gamma )^{2}/2}}{\sqrt{2\pi }} \, dt

is independent of the real parameter γ\gamma.

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