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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3answers
89 views

Evaluating an integral using Gamma function [closed]

For $r \in (0,2)$, I would like to evaluate the integral $$\frac{2}{r} \int_0^{\infty} \frac{\sin(u)}{u^r} du.$$ The answer should be $$\frac{\pi \cdot \mathrm{cosec}{\frac{r\pi}{2}} ...
0
votes
0answers
9 views

if $F(s_{0})$ for some $s_{o}$exists then it exists for all $s>s_{o}$

if laplace transform $F(s_{0})$ for some $s_{o}$exists then it exists for all $s>s_{o}$. i need to prove this . now, ...
1
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1answer
28 views

Conditions on $f(t)$ so that $\int_{-\infty}^\infty f(t) \operatorname{sinc}(t-a) \operatorname{sinc}(t-b) dt$ converges.

Let us consider $$\int_{-\infty}^\infty f(t) \operatorname{sinc}(t-a) \operatorname{sinc}(t-b) dt \ \ \ \ (*)$$ for $a,b\in \mathbb R$. If $f\in L^1(-\infty,\infty)$ the integral converges: ...
2
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2answers
52 views

How to prove and evaluate an Improper Integral

How to show that this improper integral converges and how to compute its value? $$ I=\int_{0}^{\frac\pi 2}\frac{\cos(2t)}{\sqrt{\sin(2t)}}\mathrm{d}t. $$ I used that the integrated function is odd so ...
0
votes
4answers
64 views

Evaluate $\lim _{x\to \infty }\frac{1}{x}\int _0^x\cos\left(t\right)dt\:$

I think that $\lim\limits_{x\to \infty }\frac{1}{x}\int _0^x\cos\left(t\right)dt\:$ is divergent, I can prove with taylor series?
27
votes
2answers
539 views

On the inequality $ \int_{-\infty}^{+\infty}\frac{(p'(x))^2}{(p'(x))^2+(p(x))^2}\,dx \le n^{3/2}\pi.$

$ p(x)\in\mathbb{R[X]} $ is a polynomial of degree $n$ with no real roots. Show that: $$\int\limits_{-\infty}^{+\infty}\dfrac{(p'(x))^2}{(p'(x))^2+(p(x))^2}\,dx \leq n^{3/2}\pi.$$ It's easy to ...
4
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3answers
133 views

Compute $\int_{0}^{\infty}\frac{x \log(x)}{(1+x^2)^2}dx$

Given $$\int_{0}^{\infty}\frac{x \log(x)}{(1+x^2)^2}dx$$ I couldn't evaluate this integral. My only idea here was evaluating this as integration by parts. \begin{align} \int\frac{x ...
3
votes
1answer
42 views

How to compute $\int _\mathbb{R}\frac{sin^{2n}t}{t^{2n}}dt$?

If $n=1$ we can compute $\int _\mathbb{R} \frac{sin^{2}t}{t^{2}}dt$ by using Parseval's formula since $\widehat{1_{[-1,1]}}(x)=2\frac{\sin x}{x}$. We obtain $\int _\mathbb{R} ...
2
votes
5answers
103 views

How we can solve that $\lim _{_{x\rightarrow \infty }}\int _0^x\:e^{t^2}dt$?

How we can solve that $\lim _{_{x\rightarrow \infty }}\int _0^x\:e^{t^2}dt$ ? P.S: This is my method as I thought: $\int _0^x\:\:e^{t^2}dt>\int _1^x\:e^tdt=e^x-e$ which is divergent, so all your ...
1
vote
1answer
91 views

Hints for evaluating $ \lim_{y \to +\infty}y \int_0^{+\infty}{e^{-x^2}\sin(2xy) dx}$ [closed]

Please give me some hints for this limit. $ \lim_{y \to +\infty} y\int_0^{+\infty}{e^{-x^2}\sin(2xy) dx}$
2
votes
2answers
71 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
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1answer
35 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
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0answers
44 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
84 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
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1answer
27 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
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1answer
26 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
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1answer
79 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
51 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
131 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
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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
155 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
27 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
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0answers
35 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
92 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
134 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
53 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
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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
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1answer
31 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
80 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
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0answers
23 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
519 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 ...
1
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1answer
35 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
91 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 ...
1
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1answer
51 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
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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
324 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
374 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
35 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
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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
40 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
35 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
42 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
61 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
80 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 ...