Math Problems Vault

Problem VI.1.1

(a)(b)(c)(d)

Prove that ( $p, q \in \mathbb {R}$ )

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(a)

$\int_{1}^{x} t^{p} e^{t} d t \underset {x \rightarrow +\infty }{\sim } x^{p} e^{x}$ ;

(b)

$\int_{x}^{+\infty } t^{p} e^{-t} d t \underset {x \rightarrow +\infty }{\sim } x^{p} e^{-x}$ ;

(c)

$\int_{2}^{x} t^{p-1} \ln^{q} t d t \underset {x \rightarrow +\infty }{\sim } \frac{x^{p}}{p} \ln^{q} x \quad (p>0)$ ;

(d)

$\int_{0}^{x} t^{p-1}\left|\ln t\right|^{q} d t \underset {x \rightarrow 0}{\sim } \frac{x^{p}}{p}\left|\ln x\right|^{q} \quad (p>0)$ .

Problem VI.1.2

Compute the integral $\int_{0}^{10} x^{x} d x$ with a relative error of at most 5.

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Problem VI.1.3

(a)(b)

Let $-\infty <a<b \leqslant +\infty ; f, g, h \in C([a, b))$ , with $f>0$ on $[a, b)$ and $g(t)=o(f(t))$ , $h(t) \sim f(t)$ as $t \rightarrow b-0$ . Prove that as $t \rightarrow b-0$ the following relations hold:

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(a)

if $\int_{a}^{b} f(t) d t=+\infty$ , then

\int _{a}^{x} g(t) d t=o\left(\int _{a}^{x} f(t) d t\right), \quad \int _{a}^{x} h(t) d t \sim \int _{a}^{x} f(t) d t

(b)

if $\int_{a}^{b} f(t) d t<+\infty$ , then

\int _{x}^{b} g(t) d t=o\left(\int _{x}^{b} f(t) d t\right), \quad \int _{x}^{b} h(t) d t \sim \int _{x}^{b} f(t) d t

Problem VI.1.4

(a)(b)(c)

Find the asymptotics as $t \rightarrow 1-0$ of the integrals

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(a)

$\mathscr {J}(t)=\int_{0}^{t} \frac{d x}{1-x^{a}} \quad (a \neq 0)$ ;

(b)

$\mathcal{K}(t)=\int_{0}^{1} \frac{d x}{\sqrt{\left(1-x^{2}\right)\left(1-t^{2} x^{2}\right)}}$ ;

(c)

$\mathscr {L}(t)=\int_{0}^{\pi } \frac{d x}{1-2 t \cos x+t^{2}}$ .

Problem VI.1.5

Find the asymptotics as $A \rightarrow +\infty$ of the integrals

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(a)

$\int_{A}^{2 A} \frac{d x}{\ln x}$ ;

(b)

$\int_{A}^{2 A} x^{x} d x$ ;

(c)

$\int_{0}^{A} e^{e^{-x}} d x$ ;

(d)

$\int_{\pi }^{A}\left|\sin x\right| e^{a / x} d x \quad (a \in \mathbb {R})$ ;

(e)

$\int_{A}^{+\infty } e^{-x^{2}} d x$ ;

(f)

$\int_{A}^{+\infty } x^{a} e^{-x} d x \quad (a \in \mathbb {R})$ .

Problem VI.1.6

(a)(b)(c)(d)

Find the asymptotics as $A \rightarrow +\infty$ of the integrals

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(a)

$\int_{0}^{1} e^{A x^{2}} d x$ ;

(b)

$\int_{1}^{+\infty } e^{-x^{p} / A} \frac{d x}{x} \quad (p>0)$ ;

(c)

$\int_{0}^{+\infty } e^{x-x^{2} / A} d x$ ;

(d)

$\int_{2}^{+\infty } e^{-x / A} \ln^{p} x d x \quad (p \in \mathbb {R})$ .

Problem VI.1.7

(a)(b)(c)(d)

Find the asymptotics as $n \rightarrow +\infty$ of the integrals $(n \in \mathbb {N})$

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(a)

$\quad \int_{\pi n}^{\pi (n+2)} \frac{\sin x}{x^{p}} d x \quad (p \neq 0)$ ;

(b)

$\int_{\pi n}^{+\infty } \frac{\sin x}{x^{p}} d x \quad (p>0)$ ;

(c)

$\quad \int_{\pi n}^{\pi (n+1)} \frac{\cos x}{x^{p}} d x \quad (p \neq 0)$ ;

(d)

$\int_{\pi n}^{+\infty } \frac{\cos x}{x^{p}} d x \quad (p>0)$ .

Problem VI.1.8

(a)(b)(c)(d)

Prove that as $\varepsilon \rightarrow +0$

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(a)

$\int_{1}^{+\infty } \frac{\sin^{2} x}{x^{1+\varepsilon }} d x \sim \frac{1}{2 \varepsilon }$ ;

(b)

$\int_{1}^{+\infty } \frac{\cos^{2} x}{x^{1+\varepsilon }} d x \sim \frac{1}{2 \varepsilon }$ ;

(c)

$\int_{0}^{+\infty } \frac{\sin x}{x^{\varepsilon }} d x \rightarrow 1$ ;

(d)

$\int_{0}^{+\infty } \frac{\cos x}{x^{\varepsilon }} d x \sim \frac{\pi }{2} \varepsilon$ .

Problem VI.1.9

(a)(b)(c)

Find the asymptotics as $\varepsilon \rightarrow +0$ of the integrals

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(a)

$\int_{0}^{\infty } \frac{e^{-\varepsilon x^{2}} d x}{x+\varepsilon }$ ;

(b)

$\int_{0}^{\infty } \frac{\sin \varepsilon x^{2} d x}{x^{2}+\varepsilon \sqrt{x}}$ ;

(c)

$\int_{0}^{\infty } \frac{\sin \left(x^{2}+\varepsilon \right) d x}{x^{3}+\varepsilon \sqrt{x}}$ .

Problem VI.1.10

(a)(b)

Prove that as $A \rightarrow +\infty$

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(a)

$\int_{0}^{\infty } \sin \left(x^{A}\right) d x \sim \frac{\pi }{2 A}$ ;

(b)

$\int_{0}^{\infty } \cos \left(x^{A}\right) d x \rightarrow 1$ .

Problem VI.1.11

(a)(b)

Let $C_{x}=\int_{x}^{x+1} \cos t^{2} d t$ and $S_{x}=\int_{x}^{x+1} \sin t^{2} d t$ for $x>0$ . Prove that

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(a)

$\sqrt{C_{x}^{2}+S_{x}^{2}} \leqslant \frac{1}{x}$ for all $x>0$ ;

(b)

$\sqrt{C_{x}^{2}+S_{x}^{2}}=\frac{1}{x}\left|\sin \left(x+\frac{1}{2}\right)\right|+O\left(\frac{1}{x^{2}}\right)$ as $x \rightarrow +\infty$ .

Problem VI.1.12

Let the function $\varphi$ be continuous on $\mathbb {R}$ and have period $T>0$ , with $C_{\varphi }=\frac{1}{T} \int_{0}^{T} \varphi (t) d t \neq 0$ . Prove that

\int _{1}^{+\infty } \frac{\varphi (t)}{t^{1+\varepsilon }} d t \sim \frac{C_{\varphi }}{\varepsilon } \quad \text{ as } \quad \varepsilon \rightarrow +0

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Problem VI.1.13

(a)(b)(c)(d)

Find the asymptotics as $A \rightarrow +\infty$ of the integrals

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(a)

$\int_{0}^{A} \frac{\sin^{2} x}{x^{p}} d x \quad (p \leqslant 1)$ ;

(b)

$\int_{1}^{A} \frac{\left|\sin x\right|^{p}\left|\cos x\right|}{x} d x \quad (p>-1)$ ;

(c)

$\int_{0}^{A} \frac{\left|\sin x\right|}{x^{p}} d x \quad (p \leqslant 1)$ ;

(d)

$\int_{A}^{+\infty } \frac{\left|\sin x\right|}{x^{p}} d x \quad (p>1)$ .

Problem VI.1.14

(a)(b)(c)(d)

Prove that as $A \rightarrow +\infty$

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(a)

$\int_{0}^{\pi / 2}\left(\frac{\sin A x}{\sin x}\right)^{2} d x=\frac{\pi }{2} A+O(1)$ ;

(b)

$\int_{0}^{1} \frac{\cos A x}{\sqrt{x}} d x=\sqrt{\frac{\pi }{2 A}}+O\left(\frac{1}{A}\right)$ ;

(c)

$\int_{0}^{\pi / 2} \frac{\cos^{2} x d x}{1+\cos^{2} A x} \rightarrow \frac{\pi }{4 \sqrt{2}}$ ;

(d)

$\int_{0}^{\pi / 2}\left(\frac{\sin A x}{\sin x}\right)^{4} d x=\frac{\pi }{3} A^{3}+O(A)$ .

Problem VI.1.15

Using the Poisson integral representation for the Bessel functions

J_{\nu }(A)=\frac{2}{\sqrt{\pi }} \frac{\left(\frac{A}{2}\right)^{\nu }}{\Gamma \left(\nu +\frac{1}{2}\right)} \int _{0}^{\pi / 2} \cos (A \sin \varphi ) \cos ^{2 \nu } \varphi d \varphi \quad \left(\nu >-\frac{1}{2}\right)

prove that as $\boldsymbol {A} \rightarrow +\infty$

J_{0}(A)=\frac{\sin A+\cos A}{\sqrt{\pi A}}+O\left(A^{-3 / 2}\right), \quad J_{1}(A)=\frac{\sin A-\cos A}{\sqrt{\pi A}}+O\left(A^{-3 / 2}\right)

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Problem VI.1.16

(a)(b)(c)(d)

Find the asymptotics as $A \rightarrow +\infty$ of the following integrals:

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(a)

$\int_{0}^{\infty } \frac{\cos A x d x}{x^{p}\left(1+x^{2}\right)} \quad (-2<p<1)$ ;

(b)

$\int_{0}^{\infty } \frac{\sin A x d x}{x^{p}\left(1+x^{2}\right)} \quad (\left|p\right|<2)$ ;

(c)

$\int_{0}^{\infty } \frac{\cos A x}{e^{x^{p}}} d x \quad (0<p<2)$ ;

(d)

$\int_{0}^{\infty } \frac{\sin A x}{e^{x^{p}}} d x \quad (0<p<2)$ .

Problem VI.1.17

Let the function $f$ decrease to zero on $[a,+\infty )$ , and let the function $\varphi$ be continuous on $\mathbb {R}$ and have period $T>0$ , with $\int_{0}^{T} \varphi (t) d t=0$ . Prove that

\left|\int _{A}^{+\infty } f(t) \varphi (t) d t\right| \leqslant f(A) \int _{0}^{T}\left|\varphi (t)\right| d t \quad \text{ for } \quad A \geqslant a

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Problem VI.1.18

Let the function $f$ decrease to zero on $[a,+\infty )$ , and let the function $\varphi$ be continuous on $\mathbb {R}$ and have period $T>0$ . Prove that

\int _{a}^{A} f(t) \varphi (t) d t=C_{\varphi } \int _{a}^{A} f(t) d t+I+O(f(A))

where

C_{\varphi }=\frac{1}{T} \int _{0}^{T} \varphi (t) d t, \quad I=\int _{a}^{+\infty }\left(\varphi (t)-C_{\varphi }\right) f(t) d t

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Problem VI.1.19

(a)(b)

Let the function $f$ be nonnegative and monotonic on $[a,+\infty )$ , and let the function $\varphi$ be continuous on $\mathbb {R}$ and have period $T>0$ , with $C_{\varphi }=\frac{1}{T} \int_{0}^{T} \varphi (t) d t \neq 0$ . Prove that

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(a)

if $\int_{a}^{\infty } f(t) d t=+\infty$ and $\int_{A}^{A+1} f(t) d t=o\left(\int_{a}^{A} f(t) d t\right)$ , then

\int _{a}^{A} f(t) \varphi (t) d t \sim C_{\varphi } \int _{a}^{A} f(t) d t \quad \text{ as } \quad A \rightarrow +\infty

(b)

if $\int_{a}^{\infty } f(t) d t<+\infty$ and $\int_{A}^{A+1} f(t) d t=o\left(\int_{A}^{\infty } f(t) d t\right)$ , then

\int _{A}^{+\infty } f(t) \varphi (t) d t \sim C_{\varphi } \int _{A}^{+\infty } f(t) d t \quad \text{ as } \quad A \rightarrow +\infty

Problem VI.1.20

(a)(b)

Let $f \in C(\left[a, b\right]), \varphi \in C(\mathbb {R})$ . Find the limits

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(a)

$\lim_{A \rightarrow +\infty } \int_{a}^{b} f(t) \varphi (A t) d t$ , if $\varphi$ has period $T>0$ ;

(b)

$\lim_{A \rightarrow +\infty } A \int_{a}^{b} f(t) \varphi (A t) d t$ , if $\int_{\mathbb {R}}\left|\varphi (t)\right| d t<\infty$ .

Problem VI.1.21

Let $f \in C(\left[a, b\right] \times [-1,1])$ . Prove that

I(A)=\int _{a}^{b} f(x, \sin A x) d x \underset {A \rightarrow +\infty }{\longrightarrow } \frac{1}{2 \pi } \int _{a}^{b}\left(\int _{0}^{2 \pi } f(x, \sin t) d t\right) d x

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Problem VI.1.22

(a)(b)

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(a)

Let $p(A) \geqslant 0$ and $p(A)=o\left(A^{2}\right)$ as $A \rightarrow +\infty$ . Prove that

I_{A}=\int _{0}^{\pi / 2} \frac{\sin A x}{\sin x} \cos ^{p(A)} x d x \rightarrow \frac{\pi }{2} \quad \text{ as } \quad A \rightarrow +\infty

(b)

Compute the integral

L_{A}=\int _{0}^{\pi / 2} \frac{\sin A x}{\sin x} \cos ^{A-1} x d x \quad (A>0)

Problem VI.1.23

(a)(b)

Find the asymptotics as $\varepsilon \rightarrow +0$ of the integrals

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(a)

$\int_{0}^{+\infty } e^{-x\left(1-x^{-\varepsilon }\right)} d x$ ;

(b)

$\int_{0}^{+\infty } e^{-x\left(x^{\varepsilon }-1\right)} d x$ .

Problem VI.1.24

(a)(b)(c)

Prove that as $\varepsilon \rightarrow +0$

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(a)

$\int_{0}^{1} x^{\varepsilon x} d x=1-\frac{\varepsilon }{4}+\frac{\varepsilon^{2}}{27}+O\left(\varepsilon^{3}\right)$ ;

(b)

$\int_{0}^{1} e^{-\varepsilon / \sqrt{x}} d x=1-2 \varepsilon -\varepsilon^{2} \ln \varepsilon +O\left(\varepsilon^{2}\right)$ ;

(c)

$\int_{0}^{1} \exp \left(-\frac{\varepsilon }{x \ln^{2}(x / e)}\right) d x=1-\varepsilon -\frac{\varepsilon }{\ln \varepsilon }+o\left(\frac{\varepsilon }{\ln \varepsilon }\right)$ .

Problem VI.1.25

Let the function $f$ be positive and integrable on the interval $(0,1)$ , and suppose that for some number $p$ in the interval $(1,2]$ the integral $\int_{0}^{1}\left|\ln f(x)\right|^{p} d x$ is finite. Prove that as $\varepsilon \rightarrow +0$

\int _{0}^{1} f^{\varepsilon }(x) d x=1+\varepsilon \int _{0}^{1} \ln f(x) d x+O\left(\varepsilon ^{p}\right)

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Problem VI.1.26

(a)(b)(c)(d)

Find the asymptotics as $n \rightarrow \infty$ of the following integrals:

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(a)

$\int_{0}^{\pi }\left|\sin n x\right| \frac{d x}{x}$ ;

(b)

$\int_{0}^{\pi } \max_{1 \leqslant k \leqslant n}\left|\sin k x\right| \frac{d x}{x}$ ;

(c)

$\int_{0}^{\pi } \max_{2 \leqslant k \leqslant n} \frac{\left|\sin k x\right|}{\ln k} \frac{d x}{x}$ ;

(d)

$\int_{0}^{\pi } \max_{1 \leqslant k \leqslant n} \frac{\left|\sin k x\right|}{k} \frac{d x}{x}$ .

Problem VI.1.27

Let $P(x)=a_{1} x+\ldots +a_{n} x^{n}$ be a polynomial with real coefficients, $I(P)=\int_{0}^{1} e^{i P(x)} d x$ . Prove that $\left|I(P)\right| \leqslant \frac{C_{n}}{\sqrt[n]{A}}$ , where $A=\left|a_{1}\right|+\left|a_{2}\right|+\ldots +\left|a_{n}\right|$ , and the coefficient $C_{n}$ depends only on $n$ .

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