Math Problems Vault

Problem V.1.1

Investigate the convergence of the integral $\int_{1}^{\infty } \frac{|\sin x| d x}{\exp \left(x^{2} \sin^{2} x\right)}$ .

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Problem V.1.2

Investigate the convergence of the integral $\int_{1}^{\infty } \frac{d x}{1+x^{p} \sin^{2} x}$ .

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Problem V.1.3

Investigate the convergence of the integral $\int_{1}^{\infty } \frac{x^{q} d x}{1+x^{p}|\sin x|^{r}}$ .

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Problem V.1.4

Investigate the convergence of the integral $\int_{1}^{\infty } \frac{x^{q} d x}{\exp \left(x^{p}|\sin x|^{r}\right)}$ .

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Problem V.1.5

Investigate the convergence of the integral $\int_{1}^{\infty } \frac{\left|\sin x^{p}\right|}{x^{q}} d x$ .

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Problem V.1.6

Investigate the convergence of the integral $\int_{1}^{\infty }\left|\sin \left(x^{p}+\ln^{q} x\right)\right| \frac{d x}{x}$ .

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Problem V.1.7

Investigate the convergence of the integral $\int_{1}^{\infty } x^{p}|\sin x|^{x^{q}} d x$ .

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Problem V.1.8

Investigate the convergence of the integral $\int_{1}^{\infty } \cos \left(x^{3}-x\right) d x$ .

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Problem V.1.9

Investigate the convergence of the integral $\int_{1}^{\infty } \sin (x \ln x) d x$ .

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Problem V.1.10

Investigate the convergence of the integral $\int_{1}^{\infty } \sin \left(x^{p}+a x+b\right) d x, p>1$ .

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Problem V.1.11

Investigate the convergence of the integral $\int_{\pi }^{\infty } \frac{\sin x d x}{\ln x+\cos \sqrt{x}}$ .

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Problem V.1.12

Investigate the convergence of the integral $\int_{\pi }^{\infty } \frac{\sin x d x}{\ln x+\cos x}$ .

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Problem V.1.13

Investigate the convergence of the integral $\int_{2 \pi }^{\infty } \frac{\sin x d x}{\ln x+\cos x+\sin x}$ .

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Problem V.1.14

Investigate the convergence of the integral $\int_{\pi }^{\infty } \frac{\sin x d x}{\ln x+\cos x^{2}}$ .

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Problem V.1.15

(a)(b)

Does the convergence of the integral $\int_{1}^{\infty } f(x) d x$ imply the convergence of the integrals

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(a)

$\int_{1}^{\infty } f^{3}(x) d x$ ;

(b)

$\int_{1}^{\infty }\left|f(x)\right| \frac{d x}{x^{3}}$ ?

Problem V.1.16

(a)(b)

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(a)

Prove that for a bounded function $f$ decreasing to zero as $x \rightarrow +\infty$ , the integrals

\int _{0}^{\infty }\left|\sin x\right| f(x) d x \quad \text{ and } \quad \int _{0}^{\infty } f(x) d x

converge or diverge simultaneously.

(b)

Verify that in the two-dimensional case the analogue of this statement is false:

\int _{0}^{\infty } \int _{0}^{\infty } \frac{\left|\sin x\right||\sin y|}{e^{x y} \ln (x+y+2)} d x d y<+\infty , \quad \text{ but } \quad \int _{0}^{\infty } \int _{0}^{\infty } \frac{d x d y}{e^{x y} \ln (x+y+2)}=+\infty

Problem V.1.17

Let

f(x)=\frac{1}{x^{\varepsilon }} \int _{0}^{x} \ln \left|1-t^{2}\right| \frac{d t}{t^{2}} \quad (x>0)

For which $\varepsilon >0$ does the integral $\int_{0}^{\infty } f(x) d x$ converge?

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Problem V.1.18

(a)(b)(c)

Compute the Frullani integral

\int _{0}^{\infty } \frac{f(a x)-f(b x)}{x} d x

where $a, b>0$ , the function $f$ is continuous on $[0, \infty )$ and satisfies one of the conditions:

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(a)

the integral $\int_{1}^{\infty } \frac{f(x)}{x} d x$ converges;

(b)

for some $T>0$ , for all $x \geqslant 0$ the equality $f(x+T)=f(x)$ holds;

(c)

there exists a finite limit $l=\lim_{x \rightarrow \infty } f(x)$ .

Problem V.1.19

(a)(b)

Suppose the integral $\int_{0}^{\infty } f(x) d x$ converges. Prove that

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(a)

for any $\varepsilon >0$ the integral $\int_{0}^{\infty } e^{-\varepsilon x} f(x) d x$ converges;

(b)

$\int_{0}^{\infty } e^{-\varepsilon x} f(x) d x \rightarrow \int_{0}^{\infty } f(x) d x$ as $\varepsilon \rightarrow +0$ (compare with Abel's theorem — Problem IV.5.8).

Problem V.1.20

Let $\varphi$ be a continuous $2 \pi$ -periodic function. Prove that the integral

\int _{\pi }^{\infty } \frac{\varphi (x) d x}{\ln x+\cos x}

converges only if $\varphi$ is an odd function.

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Problem V.1.21

Compute the integral $\int_{0}^{\infty } \ln \left|1-x^{2}\right| \frac{d x}{x^{2}}$ .

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Problem V.1.22

Compute the integral $\int_{0}^{\pi / 2} \ln (\sin x) d x$ .

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Problem V.1.23

Compute the integral $\int_{0}^{1} \frac{\ln x d x}{1-x}$ .

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Problem V.1.24

Compute the integral $\int_{0}^{1} \frac{\ln x d x}{1+x}$ .

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Problem V.1.25

Compute the integral $\int_{0}^{\infty } \frac{x^{2}}{\operatorname {ch}^{2} x} d x$ .

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Problem V.1.26

Compute the integral $\int_{0}^{\infty } \frac{\sin x}{x} e^{-a x} d x \quad (a>0)$ .

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Problem V.1.27

Compute the integral $\int_{0}^{\infty } \frac{\sin x}{x} d x$ .

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Problem V.1.28

Compute the integral $\int_{0}^{\infty } \frac{x-\sin x}{x^{3}} d x$ .

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Problem V.1.29

Compute the integral $\int_{0}^{\infty } \frac{e^{-x}-\cos x}{x} d x$ .

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Problem V.1.30

Compute the integral $\int_{-\infty }^{\infty } \frac{\left(e^{i a x}-1\right)\left(e^{i b x}-1\right)}{x^{2}} d x \quad (a, b \in \mathbb {R})$ .

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Problem V.1.31

Compute the integral $\int_{0}^{\infty } \frac{\ln x}{e^{x}} d x$ .

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Remark.

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The answer is expressed in terms of Euler's constant $\gamma$ (see Problem II.2.8).

Problem V.1.32

Compute the integral $\int_{0}^{\infty } \frac{\ln^{2} x}{e^{x}} d x$ .

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Remark.

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The answer is expressed in terms of Euler's constant $\gamma$ (see Problem II.2.8).

Problem V.1.33

Compute the integral $\int_{0}^{1} \frac{x^{2} d x}{(1-x) \ln^{2}(1-x)}$ .

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Problem V.1.34

Compute the integral $\int_{0}^{\infty } \frac{\ln x}{1+e^{x}} d x$ .

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Problem V.1.35

(a)(b)

Compute the integrals:

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(a)

$\int_{0}^{1} \frac{e^{-x}-1}{x} d x+\int_{1}^{\infty } \frac{e^{-x}}{x} d x$ ;

(b)

$\int_{0}^{1} \frac{\cos x-1}{x} d x+\int_{1}^{\infty } \frac{\cos x}{x} d x$ .

Remark.

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The answer is expressed in terms of Euler's constant $\gamma$ (see Problem II.2.8).

Problem V.1.36

(a)(b)

Compute the integrals:

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(a)

$\int_{0}^{\infty } \frac{\cos \left(x^{p}\right)-\cos \left(x^{q}\right)}{x} d x \quad (p, q>0)$ ;

(b)

$\int_{0}^{\infty } \frac{\exp \left(-x^{p}\right)-\exp \left(-x^{q}\right)}{x} d x \quad (p, q>0)$ .

Remark.

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The answer is expressed in terms of Euler's constant $\gamma$ (see Problem II.2.8).

Problem V.1.37

Compute the integral $\int_{1}^{\infty } \frac{\{ x\} (1-\{ x\} )}{x^{2}} d x \quad (\{ x\}$ — the fractional part of $x)$ .

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Problem V.1.38

Using the result of problem I.2.23 a), prove that

\int _{0}^{\sqrt{n}} e^{-x^{2}} d x=\sqrt{n} \frac{(2 n)!!}{(2 n+1)!!}+o(1)

Using Stirling's formula, find the value of the Euler-Poisson integral $\int_{-\infty }^{\infty } e^{-x^{2}} d x$ .

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Problem V.1.39

Compute the integral $\int_{0}^{\infty } \frac{e^{-x}-e^{-2 x}}{x^{p}} d x$ , where $p=\frac{k}{2}-1, k=0,1,2,3,4,5$ .

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Problem V.1.40

Compute the integral $\int_{0}^{\infty } \frac{e^{-x^{2}}}{\left(x^{2}+\frac{1}{2}\right)^{2}} d x$ .

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Problem V.1.41

Compute the integral $\int_{0}^{\infty } e^{-\left(x^{2}+a^{2} x^{-2}\right)} d x$ .

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Problem V.1.42

(a)(b)

Compute the integrals:

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(a)

$\int_{0}^{\infty } \sin x^{2} d x$ ;

(b)

$\int_{0}^{\infty } \cos x^{2} d x$ (Fresnel integrals).

Problem V.1.43

Compute the integral $\int_{0}^{\infty } \frac{1-|\cos x|^{p}}{x^{2}} d x \quad (p>-1)$ .

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Problem V.1.44

Compute the integral $\int_{2}^{\infty } \frac{\pi (x)}{x^{3}-x} d x$ , where $\pi (x)$ is the number of primes not exceeding $x$ .

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Problem V.1.45

Let $\varphi (x)=\frac{e^{2 \pi i x}-1}{x}, x \in \mathbb {R}$ . Prove that for any $n \in \mathbb {Z}$ , $n \neq 0$

\int _{-\infty }^{\infty } \varphi (x) \bar{\varphi }(x-n) d x=0

What is this integral equal to when $n=0$ ?

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Problem V.1.46

Prove that $\int_{0}^{\infty } x^{n} \sin (2 \pi \ln x) \frac{d x}{x^{\ln x}}=0$ for $n=0,1,2, \ldots$

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Problem V.1.47

Prove that $\int_{0}^{\pi / 2} \cos (p+1) x \cos^{p-1} x d x=0$ for all $p>0$ .

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