Questions involving improper integrals, defined as the limit of a definite integral as an endpoint of the interval of integration approaches either a specified real number or ∞ or −∞, or as both endpoints approach limits.

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2
votes
2answers
70 views

equation for the beta function

Using only the definition $$B(x, y) = \int_0^1 t^{x-1}(1-t)^{y-1}dt$$ for the Beta function, proof the term: $(x + y)B(x + 1, y) = xB(x, y) \space\space \forall x, y > 0$ . Thanks in advance! ...
2
votes
1answer
31 views

Convergence and value of improper integral

Show, that the integral $\int_0^\infty e^{-x^a}dx$ exists for all $a > 0$, and show that it's value is $\frac{1}{a}\Gamma(\frac{1}{a})$ where $\Gamma(x)$ is the gamma function. I've tried ...
1
vote
1answer
24 views

convergence of a integral using comparation

analyze the comparison criterion if the integral converges or not $\int\limits_{2}^{+\infty}\frac{\cos x}{2+e^{x^3}}dx$ attempt i used the fact that $-\frac{1}{2+e^{x^3}}\le\frac{\cos ...
1
vote
0answers
41 views

Calculate an integral with delta function

In order to calculate the integral $$ f(x) = \frac{2}{\pi}\int_0^{\pi/2}\delta\Big(x-\sqrt{1\pm\sqrt{1-\beta^2\sin^2t}}\Big)\mathrm{d}t $$ where $\beta\in(0,1]$. I am hunting for a better solution, ...
-1
votes
1answer
81 views

Integral identity involving sin(x)/x

Prove or disprove $$\displaystyle\int_{-\infty}^{\infty} \frac{3 \sin \left( x\right )}{x} \mathrm{d}x = \int_{-\infty}^{\infty} \frac{4 \sin ^ 3\left( x\right )}{x^3} \mathrm{d}x$$
1
vote
1answer
26 views

existence of an improper integral

Let $f: [1, \infty ) \to \mathbb{C}$ be a continuous function with a bounded antiderivative $F(x)$ on $[1, \infty)$. Show, that the integral $$ \int_1^\infty \frac{f(x)}{x^s} dx$$ exists for each $s ...
0
votes
1answer
20 views

Leibniz rule for an improper integral

It follows from leibniz rule that if $\frac{\partial f}{\partial \theta_0}(\theta,\theta_0)$ exists then $$\frac{d}{d\theta_0}\bigg(\int_0^{\theta_0}f(\theta,\theta_0)d\theta\bigg)=\int ...
1
vote
1answer
76 views

Which of the following is true for $\int_{1}^{0} x\ln x\, \text dx$?

Which of the following is true for $\int_{1}^{0} x\ln x\,\text dx$ it is equal to $−1/4$ it is divergent it is equal to an irrational number does not have a closed form it is impossible to ...
3
votes
1answer
49 views

Improper Integral with trigonometric functions

Determine if the following integral converges: $$\int_{-\infty}^{\infty}\frac{\cos(x)}{x^3+4x}dx.$$ So far I've thought about using the comparison test but I'm not sure how to implement it. My first ...
5
votes
1answer
114 views

Improper integral Riemann sum limit in the derivation of Fourier series to Fourier transform

To give background to my question, in all the books I've looked at to derive the inverse Fourier transform of a continuous function $f$ on $\mathbb{R}$, they seem to work as follows. Let $k$ be a ...
1
vote
1answer
27 views

proof related to convergence of a integral

i have the following condition $$0\le f(x)\le g(x)$$ and $$\int_{a}^{b}g(x)dx$$ is convergent for any $a$ and $b$ (which means $a$ or $b$ can tend to infinity) then prove that $$\int_{a}^{b}f(x)dx$$ ...
5
votes
4answers
132 views

Difficult improper integral: $\int_0^\infty \frac{x^{23}}{(5x^2+7^2)^{17}}\,\mathrm{d}x$

How can I find a closed-form expression for the following improper integral in a slick way? $$\mathcal{I}= \int_0^\infty \frac{x^{23}}{(5x^2+7^2)^{17}}\,\mathrm{d}x$$
0
votes
1answer
26 views

Integration with Respect to the Floor Function

Let $[x]$ be defined as the greatest integer part of $x \in \mathbb{R}$. Let $0<t<1$ and $\alpha(x) = [1/x]$. Compute the integral: $I(t) = \displaystyle\int_{t}^{1}x^{a}\mathrm{d\alpha(x)}$ ...
0
votes
3answers
87 views

How to evaluate the integral $\int_0^{\infty } e^{-x^2} x^4 \, dx$? [closed]

I tried to solve this expression by hand but I didn't succeed. I just have the result by integrating in Mathematica $$ \int_0^{\infty } e^{-x^2} x^4 \, dx = \frac{3 \sqrt{\pi }}{8} $$
0
votes
0answers
33 views

Numerical Triple integral with three other parameters in R

I am trying to do this triple integral $$\int_{0}^{\infty }\int_{0}^{\infty }\int_{0}^{\infty }(u+w)e^{-\frac{(u+w)^2}{2}}(v+w)e^{-\frac{(v+w)^2}{2}}(u+v)e^{-\frac{(u+v)^2}{2}}e^{-(\mu +\lambda ...
2
votes
3answers
91 views

Why does the residue method not work straight out of the box here?

I'm trying to evaluate the integral $$I = \int_0^{\infty} \frac{\cos(x)-1}{x^2}\,\mathrm{d}x $$ The way I've done this is by rewriting $\frac{\cos(x)-1}{x^2}$ as ...
7
votes
4answers
129 views

Evaluating $\int\limits_0^\infty \frac{e^x}{1+e^{2x}}\mathrm dx$, alternate methods

Problem: Evaluate $$\displaystyle\int\limits_0^\infty \frac{e^x}{1+e^{2x}}\mathrm dx$$ My progress: I have actually solved the problem, but I fear that I may not have used the "desired" methods. ...
0
votes
2answers
51 views

How to integrate $\int_{-\infty}^\infty e^{-kx^2} dx$ and $\int_{-\infty}^\infty x^2 e^{-kx^2} dx$?

Given that $$\int_{-\infty}^\infty e^{-x^2} dx = \sqrt\pi,$$ evaluate $$\int_{-\infty}^\infty e^{-kx^2} dx$$ and $$\int_{-\infty}^\infty x^2 e^{-kx^2} dx.$$ for $k>0$ I tried many approaches as ...
1
vote
4answers
59 views

Proving that the improper integral is divergent.

The task is "Evaluate the following improper integral or prove that it diverges" $$ \int_0 ^2 x^2 \ln x\,dx $$ I noticed that we can't evaluate it from $0$ to $2$, so I need to prove that it is ...
0
votes
1answer
28 views

Convergence test via integral

I've got to the problem of testing convergence using the integrals on $$ \sum _{n=1} ^{\infty} \arcsin \left( \frac{1}{\sqrt{x}} \right) $$ Our theory says: Consider an integer $N$ and a ...
3
votes
1answer
78 views

Limit of the integral $\int_0^1\frac{n\cos x}{1+x^2n^{3/2}}\,dx$

Prove that $\displaystyle\int_0^1\frac{n\cos x}{1+x^2n^{\frac32}}dx\rightarrow0$ as $n\rightarrow\infty$. $f_n(x)=\frac{n\cos x}{1+x^2n\sqrt{n}}$ tends to zero function pointwise. It just ...
1
vote
0answers
22 views

acoustic dipole volume integral w/ dirac delta?

I have an acoustic research problem that leads to the following integral formulation: \begin{align} \int_{-\infty}^{\infty}\int_{-\infty}^{\infty}p(\mathbf{y},\tau)\frac{\partial}{\partial ...
5
votes
3answers
510 views

Improper integral of a rational function!

Find the value of the integral $$\int_0^\infty \frac{x^{\frac25}}{1+x^2}dx$$ I tried the substitution $x=t^5$ to obtain $$\int_0^\infty \frac{5t^6}{1+t^{10}}dt$$ Now we can factor the denominator to ...
3
votes
2answers
56 views

Evaluate $\int_{\mathbb R} \frac{\cos[\pi(x-u)]}{1-4(x-u)^2} \frac{\cos[\pi(y-u)]}{1-4(y-u)^2} du$. [closed]

Evaluate: $$\int_{\mathbb R} \frac{\cos[\pi(x-u)]}{1-4(x-u)^2} \frac{\cos[\pi(y-u)]}{1-4(y-u)^2} du$$ where x and y are real constants. Can you help me?
1
vote
1answer
33 views

Value of $\int_{-\infty}^{\infty} \delta(t-\pi)\cos(t) \,dt$

What is the value of $$\int_{-\infty}^{\infty} \delta(t-\pi)\cos(t) \,dt?$$ I calculated the value to be infinity but I need to use the definition of the dirac delta function to prove this but I am ...
0
votes
2answers
88 views

Evaluate $\int_0^\infty \left( \frac{x^{10}}{1+x^{14}} \right)^{2} \, dx$

This is a integration question from a previous calculus exam: Evaluate $$\int_0^\infty \left( \frac{x^{10}}{1+x^{14}} \right)^{2} \, dx$$ I rewrote it as $$\lim \limits_{b \to \infty} \int_0^b ...
5
votes
2answers
153 views

$\int_{0}^{\infty}e^{-st}h(t)dt=0 \Rightarrow h(t)=0.$

Suppose $h(t)$ is continuous function and $\int_{0}^{\infty}e^{-st}h(t)dt=0 ~\forall~ s>s_{0}$, then prove that $h(t)=0$. I know "if a function is continuous, non-negative or non-positive, and its ...
0
votes
1answer
46 views

Prove $\int {\operatorname{sinc}}\big({\tau}-\lambda\big) {\operatorname{sinc}}\big({\tau}-\nu\big)d\tau= {\operatorname{sinc}}(\lambda-\nu ).$

I want to prove the following relation. For any real numbers $\lambda$ and $\nu$, we have \begin{equation} \int_{-\infty}^\infty {\operatorname{sinc}}\big({\tau}-\lambda\big) ...
3
votes
5answers
117 views

How to show $\int_0^1 \left( \frac{1}{\ln(1+x)} - \frac{1}{x} \right) \,dx$ converges?

I need to show that $$\int_0^1 \left( \frac{1}{\ln(1+x)} - \frac{1}{x} \right) \,dx$$ converges, given that $$\lim_{x\rightarrow0^+} \left( \frac{1}{\ln(1+x)} - \frac{1}{x} \right) = \frac{1}{2}$$ ...
0
votes
0answers
19 views

Dependence of finite part of integral on regularization

Recently I got stuck with some task in which integral $$ I_{\alpha \beta}(r, q) = \int \frac{d^{4}p}{(2 \pi )^{4}}\frac{p_{\alpha}(p_{\beta} + r_{\beta})}{((p - q)^{2} - m_{W}^{2})((p + q)^{2} - ...
2
votes
4answers
323 views

Where am I wrong in the following limit?

We have this function: $f(x)=\frac{2x+3}{x+2}$ and we need to find this: $$\lim _{x\to \infty \:}\frac{\int _x^{2x}f(t)\,dt}{x}$$ Now I will tell how I solved this: I suppose that $$\int _x^{2x} f(t) ...
4
votes
1answer
50 views

Counter-example to $\int_0^\infty f(x) dx=\lim_{t\to\infty} \int_{1/t}^t f(x) dx$

I want to prove or disprove the statement that, for a function $f$ that is continuous on $(0,\infty)$, we have $\displaystyle{\int_0^\infty f(x)\ dx=\lim_{t\to\infty} \int_{1/t}^t f(x)\ dx}$. My ...
5
votes
5answers
371 views

Proving that $\int_{-\pi}^{\pi} \ln |1 - e^{i\theta}| d\theta = 0$

I found this on some comprehensive exam. Prove that $\int_{-\pi}^{\pi} \ln |1 - e^{i\theta}| d\theta = 0$. I was wondering would standard approach work? By that I just mean splitting the ...
1
vote
1answer
33 views

Asymptotic behavior of the confluent hypergeometric function

Consider the following function $$U(a,z)= \frac{1}{\Gamma(a)} \int^{\infty}_0 t^{a-1} \cdot (1+t)^{-a} e^{-zt} dt$$ My Try : Let $\tau= zt$, then : $$ U(a,z)= \frac{z^{-a}}{\Gamma(a)} \int^{\infty}_0 ...
1
vote
0answers
42 views

Hints to find analytical solution to integral

I have to evaluate the expression $$f(|\vec{c}|) = \int_0^\infty \int_0^{2\pi} (z(\vec{a})+z(\vec{a}+\vec{c})) \frac{(1-\cos(\theta_{\vec{a}+\vec{c}} - ...
0
votes
1answer
37 views

An explanation of the integration

So, the integral is: $$\int_1^2\frac{x-2}{\sqrt{x-1}}dx$$ If I copied correctly from the board, the teacher said if x approaches 1+, the function approaches +$\infty$. What is the difference between ...
2
votes
1answer
36 views

How to solve this improper integral? [duplicate]

The problem is: If $f(x)\in C[0,+\infty)$, $\displaystyle\lim_{x\to+\infty}f(x)=k\in\mathbb R$, and $b>a>0$, prove: $$\int_{0}^{+\infty}\frac{f(ax)-f(bx)}{x}dx=[f(0)-k]\ln(\frac ba)$$ My ...
2
votes
1answer
33 views

Is this Brownian Integral identity correct?

$$\int_0^1 B_t dt=\lim_{\omega \to\infty}{1 \over {\omega}}{\int_0^{\omega}{Y_0+}X_t dt}$$ Where $B_t$ is simple brownian motion, and $X_t$ is a discrete random variable that can be 1 or -1 with ...
0
votes
1answer
41 views

Is there another way than linearization?

$$I= \int {\sin^mx \cos^nx }dx$$ I need a Hint on doing this integral a Successive Partial Integration but it seems that the problem shows up when $ m = 2k $ and $ n = 2p$ where $p,m \in \mathbb{N}$. ...
2
votes
4answers
60 views

Calculating value of integral of convolution using Fourier transform

Calcuate the integral $$I=\int_{-\infty}^\infty\frac{\sin a\omega\sin b\omega}{\omega\cdot \omega}d\omega.$$ First I noticed that $$\mathcal{F}(\mathbb{1}_{[-h,h]})(\omega)=\frac{\sin h ...
1
vote
2answers
79 views

Closed form of the integral $\int_0^1 \frac{x^n}{1+x}\, dx$

I am trying to evaluate the integral $$\int_0^1 \frac{x^n}{1+x}\, dx, \;\;\; n \in \mathbb{N}$$ in a closed form. I tried tackling it using Beta Form $\displaystyle \int_0^1 ...
0
votes
2answers
40 views

Convergence of the improper integral $\int_{0+}^{1-} \frac{\log x}{1-x} dx$

Let $0 < t_{1} \leq t_{2} < 1.$ Then $$\int_{t_{1}}^{t_{2}} \frac{\log x}{1-x} dx = \int_{1/t_{2}}^{1/t_{1}} \frac{\log u^{-1}}{1 - u^{-1}}(-u^{-2}) du = \int_{1/t_{2}}^{1/t_{1}} \frac{\log ...
4
votes
1answer
80 views

Prove that $ f:(a,b)\to\mathbb{R}$ is integrable iff $\lim_{\epsilon\to0} \int_{[a+\epsilon,b-\epsilon]}f$ exists

I want to solve the following: Let $ f:(a,b)\to\mathbb{R}$ continous such that $f(x)\ge 0 $ for all $x\in(a,b)$. Show that $f$ is integrable iff $\displaystyle \lim_{\epsilon\to0} ...
5
votes
3answers
123 views

Improper Integral of a periodic function converges

Given $f(x)$ is a periodic function and $\int_0^p{f(x)}dx=0$. Show $\int_1^\infty\frac{f(x)}{x}dx$ converges. 1) I know this integral can be broken into ...
0
votes
0answers
30 views

improper integral, convergent?

Fix a parameter $\theta \in (0,1/2]$. I am trying to figure out for which values of the parameter $\alpha\in \mathbb{R}$ this integral converges. $$\int_0^1 \frac{(1-(1-x)^{-\theta})^2}{x^\alpha} ...
1
vote
0answers
27 views

uniform convergence and improper integral(convolution)

I want to show that $$\frac{d}{dx}(f*g)=(\frac{d}{dx}f)*g$$ where $f(x)=\frac{1}{\sqrt{x}}e^{-\frac{1}{x}}$, $g(y)$ is continuous and bounded. the convolutions are improper integrals. I'm now here ...
7
votes
1answer
129 views

How to find the value of $I_1=\int_0^\infty\frac{\sqrt{x}\arctan{x}\log^2({1+x^2})}{1+x^2}dx$

How to find the value of $$I_1=\int_0^\infty\frac{\sqrt{x}\arctan{x}\log^2({1+x^2})}{1+x^2}dx$$ If we put $$I_2=\int_0^\infty\frac{\arctan^2({x})\log({1+x^2})}{\sqrt{x}(1+x^2)}dx$$ After long ...
0
votes
1answer
32 views

Substitution in integral, how shall I proceed

Say we have $\int_2^\infty \frac{1}{(\log n)^{\log n}}dn.$ Let $u=\log n.$ We have the boundaries become $u=\log 2$ and $u=\infty.$ How should I proceed with $dn.$ I have $du=\frac{1}{n}dn,$ hence, ...
0
votes
1answer
35 views

Transform an Integral bounds from -inf, inf to 0 to 1

Good day, If i have an integral from -infinity to infinity, how do I change the bounds/limits to 0 to 1? I don't want to give exact question since this is part of an assignment. I know how to figure ...
4
votes
0answers
42 views

Integral involving modified Bessel function of the second kind

I would like to calculate the closed-form expression for the following integral: $$ I = \int_{0}^{\infty} x^{M}\exp(-\frac{x}{a})K_{\nu}(b\sqrt{1+x})\mathrm{d}x,$$ where $M$, $a$, and $b$ are all ...