Math Problems Vault

Problem 3.13.1

(1)(2)

Prove that the function $I(\alpha )$ is continuous on $R$ , if:

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(1)

$I(\alpha ) = \int_{0}^{1} \sin^{2} \alpha x^{2} d x$ ;

(2)

$I(\alpha ) = \int_{-1}^{2} \frac{x^{2}}{1+x^{2}+\alpha^{2} x^{4}} d x$ .

Problem 3.13.2

Find the limit:

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(1)

$\lim_{\alpha \rightarrow 0} \int_{0}^{1} \sqrt{1+\alpha^{2} x^{4}} d x$ ;

(2)

$\lim_{\alpha \rightarrow 0} \int_{-1}^{1} \sqrt{x^{2}+\alpha^{2}} d x$ ;

(3)

$\lim_{\alpha \rightarrow 1} \int_{2}^{4} \frac{x d x}{1+x^{2}+\alpha^{6}}$ ;

(4)

$\lim_{\alpha \rightarrow 1} \int_{0}^{1} x^{2} e^{\alpha x^{3}} d x$ ;

(5)

$\lim_{\alpha \rightarrow 0} \int_{0}^{\pi } x \cos (1+\alpha ) x d x$ . on $R$ .

Problem 3.13.3

Prove that the function $I(\alpha ) = \int_{0}^{1} \operatorname {sign}(x-\alpha ) d x$ is continuous $R$ .

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Problem 3.13.4

Let the function $f(x)$ be continuous and take positive values on the segment $[0 ; 1]$ . Prove that the function

I(\alpha ) = \int _{0}^{1} \frac{\alpha }{x^{2}+\alpha ^{2}} f(x) d x

is discontinuous at $\alpha = 0$ .

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Problem 3.13.5

(1)(2)

Determine whether the equality holds

\lim _{\alpha \rightarrow 0} \int _{0}^{1} f(x ; \alpha ) d x = \int _{0}^{1} \lim _{\alpha \rightarrow 0} f(x ; \alpha ) d x

if:

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(1)

$f(x ; \alpha ) = \frac{x}{\alpha^{2}} e^{-x^{2} / a^{2}}$ ;

(2)

$f(x ; \alpha ) = \frac{2 x a^{2}}{\left(\alpha^{2}+x^{2}\right)^{2}}$ .

Problem 3.13.6

Let the function $f(x)$ be continuous on the segment $[a ; b]$ and $a < a_{0} < x < b$ . Prove that

\lim _{\alpha \rightarrow 0} \frac{1}{\alpha } \int _{\alpha _{0}}^{x}[f(t+\alpha )-f(t)] d t = f(x)-f\left(a_{0}\right)

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Problem 3.13.7

Let $\left\{ \varphi_{n}(x)\right\}$ be a sequence of functions, Riemann integrable on the segment $[a ; b]$ and taking nonnegative values on this segment, converging uniformly to zero on the set $E = \left\{ x: 0 < \delta \leqslant \left|x\right| \leqslant 1\right\}$ for every $\delta > 0$ , and moreover

\lim _{n \rightarrow \infty } \int _{-1}^{1} \varphi _{n}(x) d x = 1

Prove that for any function $f(x)$ continuous on the segment $[-1 ; 1]$ , the equality holds

\lim _{n \rightarrow \infty } \int _{-1}^{1} f(x) \varphi _{n}(x) d x = f(0)

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Problem 3.13.8

(1)(2)(3)

Determine whether the integrals are equal

\int _{0}^{1}\left(\int _{0}^{1} f(x ; \alpha ) d \alpha \right) d x \quad \text{ and } \quad \int _{0}^{1}\left(\int _{0}^{1} f(x ; \alpha ) d x\right) d \alpha

if:

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(1)

$f(x ; \alpha ) = \frac{\alpha^{2}-x^{2}}{\left(\alpha^{2}+x^{2}\right)^{2}}$ ;

(2)

$f(x ; \alpha ) = \frac{\alpha -x}{(\alpha +x)^{3}}$ ;

(3)

$f(x ; \alpha ) = \left(\frac{x^{5}}{\alpha^{4}}-\frac{2 x^{3}}{\alpha^{3}}\right) e^{-x^{2} / \alpha }$ .

Problem 3.13.9

(1)(2)

Let the function $f(x ; \alpha )$ , for each $\alpha \in \left[\alpha_{1} ; \alpha_{2}\right]$ , be integrable in $x$ on the segment $[a ; b]$ , and suppose on this segment there exists a function $\varphi (x)$ such that $\lim_{\alpha \rightarrow \alpha_{0}} f(x ; \alpha ) = \varphi (x)$ , where $\alpha_{0} \in \left[\alpha_{1} ; \alpha_{2}\right]$ , uniformly with respect to $x \in [a ; b]$ . Prove that:

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(1)

$\lim_{\alpha \rightarrow \alpha_{0}} \int_{a}^{b} f(x ; \alpha ) d x = \int_{a}^{b} \varphi (x) d x$ ;

(2)

$\lim_{\alpha \rightarrow \alpha_{0}} \int_{a}^{b} f(x ; \alpha ) g(x) d x = \int_{a}^{b} \varphi (x) g(x) d x$ , where $g(x)$ is a function integrable on the segment $[a ; b]$ .

Problem 3.13.10

Using the formula $\frac{\operatorname {arctg} x}{x} = \int_{0}^{1} \frac{d \alpha }{1+x^{2} \alpha^{2}}$ , calculate the integral

\int _{0}^{1} \frac{\operatorname {arctg} x}{x \sqrt{1-x^{2}}} d x

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Problem 3.13.11

Using the formula

\frac{1}{\sin x} \ln \frac{a+b \sin x}{a-b \sin x} = 2 a b \int _{0}^{1} \frac{d t}{a^{2}-b^{2} t^{2} \sin ^{2} x}

where $a > b > 0$ , calculate the integral

\int _{0}^{\pi / 2} \ln \frac{a+b \sin x}{a-b \sin x} \frac{d x}{\sin x}

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Problem 3.13.12

(1)(2)

Let $a > 0, b > 0$ . Calculate the integral:

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(1)

$\int_{0}^{1} \sin \left(\ln \frac{1}{x}\right) \frac{x^{b}-x^{a}}{\ln x} d x$ ;

(2)

$\int_{0}^{1} \cos \left(\ln \frac{1}{x}\right) \frac{x^{b}-x^{a}}{\ln x} d x$ .

Problem 3.13.13

(1)(2)(3)(4)

Find $I^{\prime }(\alpha )$ , if:

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(1)

$I(\alpha ) = \int_{0}^{1} \sin (\alpha x) d x ;$

(2)

$I(\alpha ) = \int_{1}^{3} \frac{\cos \left(\alpha x^{3}\right)}{x} d x$ ;

(3)

$I(\alpha ) = \int_{1}^{2} e^{\alpha x^{2}} \frac{d x}{x}$ ;

(4)

$I(\alpha ) = \int_{2}^{3} \operatorname {ch}\left(\alpha^{4} x^{2}\right) \frac{d x}{x}$ .

Problem 3.13.14

Find $\Phi^{\prime }(\alpha )$ , if:

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(1)

$\Phi (\alpha ) = \int_{0}^{\alpha } \frac{\ln (1+\alpha x)}{x} d x$ ;

(2)

$\Phi (\alpha ) = \int_{\alpha }^{2 \alpha } \frac{\sin \alpha x}{x} d x$ ;

(3)

$\Phi (\alpha ) = \int_{\sin \alpha }^{\cos \alpha } e^{\alpha \sqrt{1-x^{2}}} d x$ ;

(4)

$\Phi (\alpha ) = \int_{3 \alpha }^{\alpha^{2}} e^{\alpha x^{2}} d x$ ;

(5)

$\Phi (\alpha ) = \int_{\cos \alpha }^{\sin \alpha } e^{\alpha^{4} x^{2}} d x$ ;

(6)

$\Phi (\alpha ) = \int_{e^{-\alpha }}^{e^{\alpha }} \ln \left(1+\alpha^{2} x^{2}\right) \frac{d x}{x}$ ;

(7)

$\Phi (\alpha ) = \int_{\alpha e^{-\alpha }}^{\alpha e^{\alpha }} \ln \left(1+\alpha^{2} x^{2}\right) d x$ ;

(8)

$\Phi (\alpha ) = \int_{\operatorname {ch} \alpha }^{\operatorname {sh} \alpha } \ln \left(1+x^{2}+\alpha^{2}\right) d x$ .

Problem 3.13.15

Can the derivative of the function be calculated using Leibniz's rule

I(\alpha ) = \int _{0}^{1} \ln \left(x^{2}+\alpha ^{2}\right) d x \quad \text{ at } \quad \alpha = 0?

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Problem 3.13.16

Let the function $f$ be continuous on $R$ . Prove that the function $F(x) = \frac{1}{2 a} \int_{-a}^{a} f(x+t) d t$ , where $a > 0$ , has a continuous derivative on $R$ , and find $F^{\prime }(x)$ .

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Problem 3.13.17

By differentiating the integral $\int_{0}^{b} \frac{d x}{x^{2}+\alpha^{2}}$ with respect to the parameter $\alpha$ , where $\alpha > 0$ , calculate the integral $\int_{0}^{b} \frac{d x^{0}}{\left(x^{2}+\alpha^{2}\right)^{2}}$ .

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Problem 3.13.18

(1)(2)(3)(4)

Using differentiation with respect to the parameter $\alpha$ , calculate the integral $I(\alpha )$ , if:

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(1)

$I(\alpha ) = \int_{0}^{\pi / 2} \ln \left(\alpha^{2}-\sin^{2} \varphi \right) d \varphi , \quad \alpha > 1$

(2)

$I(\alpha ) = \int_{0}^{\pi } \ln \left(1-2 \alpha \cos x+\alpha^{2}\right) d x,\left|\alpha \right| < 1$

(3)

$I(\alpha ) = \int_{0}^{\pi } \ln \frac{1+\alpha \cos x}{1-\alpha \cos x} \frac{d x}{\cos x},\left|\alpha \right| < 1$

(4)

$I(\alpha ) = \int_{0}^{\pi / 2}-\operatorname {arctg}(\alpha \operatorname {tg} x) \cdot d x$ .

Problem 3.13.19

(1)(2)(3)

Let the function $f(x ; \alpha )$ be continuous in the rectangle

K = \left\{ (x ; \alpha ): a \leqslant x \leqslant b, \alpha _{1} \leqslant \alpha \leqslant \alpha _{2}\right\}

and the function $g(x)$ is integrable on the segment $[a ; b]$ . Prove that:

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(1)

the function $F(\alpha ) = \int_{a}^{b} f(x ; \alpha ) g(x) d x$ is continuous on the segment $\left[\alpha_{1} ; \alpha_{2}\right] ;$

(2)

$\int_{\alpha_{1}}^{\alpha_{2}} F(\alpha ) d \alpha = \int_{a}^{b}\left(\int_{\alpha_{1}}^{\alpha_{2}} f(x ; \alpha ) g(x) d \alpha \right) d x$

(3)

the function $F(\alpha )$ is continuously differentiable on the segment $\left[\alpha_{1} ; \alpha_{2}\right]$ , and moreover

F^{\prime }(\alpha ) = \int _{a}^{b} \frac{\partial f(x ; \alpha )}{\partial \alpha } g(x) d x

under the additional condition that the function $\frac{\partial f(x ; \alpha )}{\partial \alpha }$ is continuous in the rectangle $K$ .

Problem 3.13.20

Let $F(\alpha ) = \int_{0}^{\alpha }(x+\alpha ) f(x) d x$ , where $f(x)$ is a function differentiable on $R$ . Find $F^{\prime \prime }(a)$ .

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Problem 3.13.21

Let $F(a) = \int_{a}^{b} f(x)\left|x-\alpha \right| d x$ , where $f(x)$ is a function continuous on the segment $[a ; b]$ . Find $F^{\prime \prime }(\alpha )$ .

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Problem 3.13.22

Let $F(\alpha ) = \frac{1}{h^{2}} \int_{0}^{h}\left(\int_{0}^{h} h(\xi +\eta +\alpha ) d \eta \right) d \xi$ , where $h > 0, f$ a function continuous on $R$ . Find $F^{\prime \prime }(\alpha )$ .

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Problem 3.13.23

Let the function $f$ be continuous on the segment $[a ; b], x_{0} \in (a ; b), x \in (a ; b), k \neq 0$ . Prove that the function

y(x) = \frac{1}{k} \int _{x_{0}}^{x} f(t) \sin k(x-t) d t

satisfies the differential equation $y^{\prime \prime }+k^{2} y = f(x)$ .

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Problem 3.13.24

Let the function $f$ be continuous on the segment $[a ; b], x_{0} \in (a ; b), x \in (a ; b)$ . Prove that the function

F(x) = \frac{1}{(n-1) !} \int _{x_{0}}^{x}(x-t)^{n-1} f(t) d t, \quad \text{ where } \quad n \in N

satisfies the conditions

F\left(x_{0}\right) = F^{\prime }\left(x_{0}\right) = \ldots = F^{(n-1)}\left(x_{0}\right) = 0, \quad F^{(n)}(x) = f(x)

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Problem 3.13.25

Prove that the function

u(r) = \int _{0}^{\pi } e^{n r \cos \theta } d \theta

for any $n \in Z$ satisfies the equation

\frac{d^{2} u}{d r^{2}}+\frac{1}{r} \frac{d u}{d r}-n^{2} u = 0

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Problem 3.13.26

(1)(2)

Prove that the function $u(x)$ satisfies Bessel's equation

x^{2} u^{\prime \prime }+x u^{\prime }+\left(x^{2}-n^{2}\right) u = 0

if:

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(1)

$u(x) = \frac{1}{\pi } \int_{0}^{\pi } \cos (n \varphi -x \sin \varphi ) d \varphi , \quad n \in N$

(2)

$u(x) = x^{n} \int_{0}^{\pi } \cos (x \cos \varphi ) \sin^{2 n} \varphi d \varphi , \quad n \in N$ .

Problem 3.13.27

(1)(2)(3)(4)

Consider the complete elliptic integrals

E(k) = \int _{0}^{\pi / 2} \sqrt{1-k^{2} \sin ^{2} \varphi } d \varphi , \quad K(k) \equiv F(k) = \int _{0}^{\pi / 2} \frac{d \varphi }{\sqrt{1-k^{2} \sin ^{2} \varphi }},

where $0 < k < 1$ . Prove that:

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(1)

$E^{\prime }(k) = \dfrac {E(k)-K(k)}{k}, \quad K^{\prime }(k) = \dfrac {E(k)}{k\left(1-k^{2}\right)}-\dfrac {K(k)}{k}$ ;

(2)

$E^{\prime \prime }(k)+\frac{1}{k} E^{\prime }(k)+\frac{E(k)}{1-k^{2}} = 0$ ;

(3)

$\int_{0}^{k} t K(t) d t = E(k)-\left(1-k^{2}\right) K(k)$ ;

(4)

$\int_{0}^{k} t E(t) d t = \frac{1}{3}\left(\left(1+k^{2}\right) E(k)-\left(1-k^{2}\right) K(k)\right)$ .

Problem 3.13.28

(1)(2)

Let $\varphi (x) = \frac{\sin x}{x}$ for $x \neq 0, \varphi (0) = 1$ . Show that:

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(1)

$x^{n+1} \varphi^{(n)}(x) = \int_{0}^{x} t^{n} \cos \left(t+\frac{n \pi }{2}\right) d t, n \in N, x \in R$ ;

(2)

$\left|\varphi^{(n)}(x)\right| \leqslant \frac{1}{n+1}, n \in N, x \in R$ .

Problem 3.13.29

Let the function $\varphi (x)$ and its derivative $\varphi^{\prime }(x)$ be continuous on the segment $[0 ; a]$ , and let

F(t) = \int _{0}^{t} \frac{\varphi (x) d x}{\sqrt{t-x}}

Prove that for $t \in (0 ; a)$ the equality holds

F^{\prime }(t) = \frac{\varphi (0)}{\sqrt{t}}+\int _{0}^{t} \frac{\varphi ^{\prime }(x)}{\sqrt{t-x}} d x

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Problem 3.13.30

Let $K(x ; y) = \begin{cases} x(1-y), & \text{ if } x \leqslant y, \\ y(1-x), & \text{ if } x > y, \end{cases}$ and let $\varphi (y)$ be a function continuous on the segment $[0 ; 1]$ . Show that the function

u(x) = \int _{0}^{1} K(x ; y) \varphi (y) d y

satisfies on the segment $[0 ; 1]$ the equation $u^{\prime \prime }(x) = -\varphi (x)$ .

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Problem 3.13.31

(1)(2)(3)

Find a function $\varphi (x)$ , twice differentiable on $R$ , satisfying the equation:

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(1)

$\varphi (x) = x+\int_{0}^{x}(y-x) \varphi (y) d y$

(2)

$\varphi (x) = 1+\lambda \int_{0}^{x}(x-y) \varphi (y) d y, \quad \lambda > 0$ ;

(3)

$\varphi (x) = \lambda \int_{0}^{x}(x-y) \varphi (y) d y+x^{2}, \quad \lambda > 0$ .

Problem 3.13.32

Find $F_{x y}^{\prime \prime }(x ; y)$ , if $F(x ; y) = \int_{x / y}^{x y}(x-y t) f(t) d t$ , where $f(t)$ is a differentiable function, $y \neq 0$ .

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Problem 3.13.33

Let $f$ be a twice differentiable function and $F$ a differentiable function. Prove that the function

u(x ; t) = \frac{1}{2}(f(x-a t)+f(x+a t))+\frac{1}{2 a} \int _{x-a t}^{x+a t} F(\xi ) d \xi

satisfies the wave equation

\frac{\partial ^{2} u}{\partial t^{2}} = a^{2} \frac{\partial ^{2} u}{\partial x^{2}}, \quad a \neq 0

and the following initial conditions:

u(x ; 0) = f(x), \quad u_{t}^{\prime }(x ; 0) = F(x)

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Problem 3.13.34

Show that if the function $f$ is continuous on the segment $[0 ; a]$ , $\xi \in [0 ; a]$ and $(x-\xi )^{2}+y^{2}+z^{2} \neq 0$ , then the function

u(x ; y ; z) = \int _{0}^{a} \frac{f(\xi ) d \xi }{\sqrt{(x-\xi )^{2}+y^{2}+z^{2}}}

satisfies Laplace's equation $\frac{\partial^{2} u}{\partial x^{2}}+\frac{\partial^{2} u}{\partial y^{2}}+\frac{\partial^{2} u}{\partial z^{2}} = 0$ .

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