Math Problems Vault

Problem 3711

Prove that the integral

F(y)=\int _{0}^{1}f(x,y)\, dx

of the discontinuous function $f(x,y)=\operatorname {sgn}(x-y)$ is a continuous function. Plot the graph of the function $u=F(y)$ .

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Problem 3712

Investigate the continuity of the function

F(y)=\int _{0}^{1}\dfrac {yf(x)}{x^{2}+y^{2}}\, dx

where $f(x)$ is a positive continuous function on the closed interval $\left[0,1\right]$ .

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Problem 3713

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

Find:

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(a)

(b)

$\lim \limits_{a \to 0}\displaystyle \int_{-1}^{1}\sqrt{x^{2}+a^{2}}\, dx$ ;

(c)

$\lim \limits_{a \to 0}\displaystyle \int_{0}^{2}x^{2}\cos ax\, dx$ ;

(d)

$\lim \limits_{n \to \infty }\displaystyle \int_{0}^{1}\dfrac {dx}{1+\left(1+\dfrac {x}{n}\right)^{n}}$ .

Problem 3713.1

$\lim \limits_{a \to 0}\displaystyle \int_{a}^{1+a}\dfrac {dx}{1+x^{2}+a^{2}}$ ;

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Problem 3714

Let the function $f(x)$ be continuous on the closed interval $\left[a,A\right]$ . Prove that

\lim _{h \to +0}\dfrac {1}{h}\int _{a}^{x}\left[f(t+h)-f(t)\right]\, dt=f(x)-f(a) \quad (a<x<A) .

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Problem 3714.1

[NO TASK]

Problem 3715

In the expression below, can the limit operation be carried out under the integral sign

\lim _{y \to 0}\int _{0}^{1}\dfrac {x}{y^{2}}e^{-\frac{x^{2}}{y^{2}}}\, dx \, ?

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Problem 3716

When $y=0$ , can Leibniz's rule be used to compute the derivative of the function

F(y)=\int _{0}^{1}\ln \sqrt{x^{2}+y^{2}}\, dx

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Problem 3717

If

F(x)=\int _{x}^{x^{2}}e^{-xy^{2}}\, dy ,

compute $F'(x)$ .

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Problem 3718

Let:

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(a)

$F(a)=\displaystyle \int_{\sin a}^{\cos a}e^{a\sqrt{1-x^{2}}}\, dx$ ;

(b)

$F(a)=\displaystyle \int_{a+\alpha }^{b+\alpha }\dfrac {\sin ax}{x}\, dx$ ;

(c)

$F(a)=\displaystyle \int_{0}^{a}\dfrac {\ln (1+ax)}{x}\, dx$ ;

(d)

$F(a)=\displaystyle \int_{0}^{a}f(x+a,x-a)\, dx$ ;

(e)

$F(a)=\displaystyle \int_{0}^{a^{2}}dx\int_{x-a}^{x+a}\sin (x^{2}+y^{2}-a^{2})\, dy$ ,

find $F'(a)$ .

Problem 3719

If

F(x)=\int _{0}^{x}(x+y)f(y)\, dy ,

where $f(x)$ is a differentiable function, find $F''(x)$ .

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Problem 3720

Let:

F(x)=\int _{a}^{b}f(y)\left|x-y\right|\, dy ,

where $a<b$ and $f(y)$ is a differentiable function, find $F''(x)$ .

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Problem 3721

Let:

F(x)=\dfrac {1}{h^{2}}\int _{0}^{h}d\xi \int _{0}^{h}f(x+\xi +\eta )\, d\eta \quad (h>0) ,

where $f(x)$ is a continuous function, find $F''(x)$ .

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Problem 3722

Let:

F(x)=\int _{0}^{x}f(t)(x-t)^{n-1}\, dt ,

find $F^{(n)}(x)$ .

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Problem 3722.1

[NO TASK]

Problem 3723

On the interval $1 \leqslant x \leqslant 3$ , approximate the function $f(x)=x^{2}$ by the linear function $a+bx$ so that

\int _{1}^{3}(a+bx-x^{2})^{2}\, dx=\min .

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Problem 3724

Given the condition that the mean square deviation between the functions $a+bx$ and $\sqrt{1+x^{2}}$ on the given interval $\left[0,1\right]$ is minimal, find the approximate formula

\sqrt{1+x^{2}} \approx a+bx \quad (0 \leqslant x \leqslant 1) .

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Problem 3725

Find the complete elliptic integrals

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

and

F(k)=\int _{0}^{\frac{\pi }{2}}\dfrac {d\varphi }{\sqrt{1-k^{2}\sin ^{2}\varphi }} \quad (0<k<1)

and express their derivatives in terms of the functions $E(k)$ and $F(k)$ .

Prove that $E(k)$ satisfies the differential equation

E''(k)+\dfrac {1}{k}E'(k)+\dfrac {E(k)}{1-k^{2}}=0 .

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Problem 3726

Prove that: the Bessel function of integer order $n$

J_{n}(x)=\dfrac {1}{\pi }\int _{0}^{\pi }\cos (n\varphi -x\sin \varphi )\, d\varphi

satisfies Bessel's equation

x^{2}J''_{n}(x)+xJ'_{n}(x)+(x^{2}-n^{2})J_{n}(x)=0 .

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Problem 3727

Let:

I(a)=\int _{0}^{a}\dfrac {\varphi (x)\, dx}{\sqrt{a-x}} ,

where the function $\varphi (x)$ and its derivative $\varphi '(x)$ are continuous on the closed interval $0 \leqslant x \leqslant a$ .

Prove that when $0<a<\alpha$ ,

I'(a)=\dfrac {\varphi (0)}{\sqrt{a}}+\int _{0}^{a}\dfrac {\varphi ’(x)}{\sqrt{a-x}}\, dx .

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Problem 3728

Given the function

u(x)=\int _{0}^{1}K(x,y)v(y)\, dy ,

where

K(x,y)= \begin{cases} x(1-y) , & \text{if } x \leqslant y; \\ y(1-x) , & \text{if } x \geqslant y, \end{cases}

and $v(y)$ are both continuous. Prove that the given function satisfies the equation

u''(x)=-v(x) \quad (0 \leqslant x \leqslant 1) .

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Problem 3729

Let:

F(x,y)=\int _{\frac{x}{y}}^{xy}(x-yz)f(z)\, dz ,

where $f(z)$ is a differentiable function, find $F''_{xy}(x,y)$ .

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Problem 3730

Let $f(x)$ be a twice-differentiable function and $F(x)$ a differentiable function. Prove that the function

u(x,t)=\dfrac {1}{2}\left[f(x-at)+f(x+at)\right]+\dfrac {1}{2a}\int _{x-at}^{x+at}F(z)\, dz

satisfies the wave equation for a vibrating string

\dfrac {\partial ^{2}u}{\partial t^{2}}=a^{2}\dfrac {\partial ^{2}u}{\partial x^{2}}

together with the initial conditions: $u(x,0)=f(x)$ , $u'_{t}(x,0)=F(x)$ .

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Problem 3731

Prove that: if the function $f(x)$ is continuous on the closed interval $\left[0,l\right]$ and $(x-\xi )^{2}+y^{2}+z^{2} \neq 0$ when $0 \leqslant \xi \leqslant l$ , then the function

u(x,y,z)=\int _{0}^{l}\dfrac {f(\xi )\, d\xi }{\sqrt{(x-\xi )^{2}+y^{2}+z^{2}}}

satisfies Laplace's equation

\dfrac {\partial ^{2}u}{\partial x^{2}}+\dfrac {\partial ^{2}u}{\partial y^{2}}+\dfrac {\partial ^{2}u}{\partial z^{2}}=0 .

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Problem 3732

Using differentiation with respect to a parameter, evaluate the integral

\int _{0}^{\frac{\pi }{2}}\ln (a^{2}\sin ^{2}x+b^{2}\cos ^{2}x)\, dx .

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Problem 3733

Using differentiation with respect to a parameter, evaluate the integral

\int _{0}^{\pi }\ln (1-2a\cos x+a^{2})\, dx .

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Problem 3734

Using differentiation with respect to a parameter, compute the integral

\int _{0}^{\frac{\pi }{2}}\dfrac {\operatorname {arctg}(a\operatorname {tg}x)}{\operatorname {tg}x}\, dx .

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Problem 3735

Using differentiation with respect to a parameter, compute the integral

\int _{0}^{\frac{\pi }{2}}\ln \dfrac {1+a\cos x}{1-a\cos x}\cdot \dfrac {dx}{\cos x} \quad (\left|a\right|<1) .

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Problem 3736

Using the formula

\dfrac {\operatorname {arctg}x}{x}=\int _{0}^{1}\dfrac {dy}{1+x^{2}y^{2}} ,

compute the integral $\displaystyle \int_{0}^{1}\dfrac {\operatorname {arctg}x}{x}\cdot \dfrac {dx}{\sqrt{1-x^{2}}}$ .

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Problem 3737

Using integration under the integral sign, compute the integral

\int _{0}^{1}\dfrac {x^{b}-x^{a}}{\ln x}\, dx \quad (a>0,\ b>0) .

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Problem 3738

(a)(b)

Compute the integrals:

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(a)

$\displaystyle \int_{0}^{1}\sin \left(\ln \dfrac {1}{x}\right)\dfrac {x^{b}-x^{a}}{\ln x}\, dx$ ;

(b)

$\displaystyle \int_{0}^{1}\cos \left(\ln \dfrac {1}{x}\right)\dfrac {x^{b}-x^{a}}{\ln x}\, dx \quad (a>0,\ b>0)$ .

Problem 3739

(a)(b)

Let $F(k)$ and $E(k)$ be complete elliptic integrals (see Problem 3725). Prove the formulas:

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(a)

$\displaystyle \int_{0}^{k}F(k)k\, dk=E(k)-k_{1}^{2}F(k)$ ;

(b)

$\displaystyle \int_{0}^{k}E(k)k\, dk=\dfrac {1}{3}\left[(1+k^{2})E(k)-k_{1}^{2}F(k)\right]$ ,

where $k_{1}^{2}=1-k^{2}$ .

Problem 3740

Prove the formula

\int _{0}^{x}xJ_{0}(x)\, dx=xJ_{1}(x) ,

where $J_{0}(x)$ and $J_{1}(x)$ are the Bessel functions of order $0$ and $1$ respectively (see Problem 3726).

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