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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0
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0answers
31 views

Applying contour integration to $\int_{0}^{\pi}dx\frac{cos(x)}{\sqrt{x^2 + x_0^2}}$

Is it possible to apply contour integration to find the value of following integral $$\int_{0}^{\pi}dx\frac{cos(x)}{\sqrt{x^2 + x_0^2}}$$
0
votes
0answers
29 views

Integral of $\frac{\exp\left(\, -\alpha x\,\right)\, (x-x_0)} {{(x-x_0)^2+\beta^2}}$

Does the following integral have a closed form solution? $$ \int_{0}^{\infty} \frac{\exp\left(\, -\alpha x\,\right)\, (x-x_0)} {{(x-x_0)^2+\beta^2}}{\rm d}x $$ where $\alpha$, $\beta$ and $x_0$ are ...
2
votes
1answer
19 views

Find the inverse fourier transform of simple function

Suppose that the fourier transform of a signal $x(t)$ is $\hat x(u)=\frac{1}{2u_m}(1+\cos (\frac{\pi u}{u_m}))$ where $u_m \geq |u|$.$t$ here stands for time so $t \geq 0$ We sample the original ...
0
votes
2answers
18 views

Improper integrals using comparison theorem

In their working out I understand for the numerator 2+cosx=3 as cosx is less than equal to 1 but in the denominator I don't understand how they got from 3 square root x-x squared sinx all the way ...
2
votes
2answers
63 views

What is the largest function whose integral still converges?

Let C be the set of all functions $f(x)$ whose integral converges, i.e. for some constant $x_0$: $$\int_{x_0}^\infty f(x) dx < \infty$$ While playing with integrals in Wolfram Alpha, I noticed ...
0
votes
0answers
23 views

Dirichlet integral using real Analysis

The teacher made this approach to solve the Dirichlet integral , $$ J_n= \int_0^\frac{\pi}{2} \frac{\sin(2nx)}{\sin x}\:\mathrm{d}x,\quad I_n = \int_0^\frac{\pi}{2} \frac{\sin(2n+1)x}{\sin ...
2
votes
3answers
73 views

Compute the integral $\int_{0}^{\infty} \frac{(1 + x + x^2)}{(1+x^4)} dx $ with a residue on suitable contour.

I believe that I could try to compute the same integral with limits from $-\infty$ to $\infty$ using residue on a half circle and then let the radius tend off to infinity, and once I have that value I ...
1
vote
1answer
58 views

Does the improper integral $\int_{0}^{\infty}\sin(x^2)\;\mathrm dx$ converge? [duplicate]

Does the improper integral $\int_{0}^{\infty}\sin(x^2)\;\mathrm dx$ converge? So if it converges then $\lim_{b \to\infty}\int_{0}^{b}\sin(x^2)\;\mathrm dx$ exists and our integral converges to this ...
0
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1answer
35 views

Calculating Fourier Transform of $\sum_{n=1}^{3}\sin(2\pi \frac{n}{8}\frac{t}{T})$

This question deals with finding the Nyquist Frequency of a given signal. Suppose you have the signal $x(t)=\sum_{n=1}^{3}\sin(2\pi \frac{n}{8}\frac{t}{T})$ in the time domain where $T>0$ is some ...
-1
votes
0answers
17 views

How to evaluate the area of the curve?

I want to evaluate the area of the following between $x=0$ and $x=1$: $$f(x)=\frac{\log(1+x)}{1+x^2}$$ I started by introducing a suitable trigonometric substitution but according to what I did I ...
1
vote
3answers
74 views

Evaluating an integral using Gamma function [on hold]

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
8 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
vote
1answer
27 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
votes
3answers
44 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
59 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?
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0answers
135 views
+50

Integral Inequality $\leq n^{3/2}\pi$

$ 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 see ...
-4
votes
0answers
40 views

what is the result of an infinity minus pi [closed]

what is $ \infty- \pi? $ Im doing some questions related to improper integral but then the final answer I got is $\tan^{-1} (\infty) - \tan^{-1} (\pi) .$ So what should I get as the final answer ?
4
votes
3answers
117 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
37 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
97 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
87 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
69 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
29 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
23 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
40 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
74 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
44 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
93 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
130 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? $$\mathcal{I}= \int_0^\infty \frac{x^{23}}{(5x^2+7^2)^{17}}\,\mathrm{d}x$$
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votes
1answer
25 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
81 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
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0answers
23 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
89 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
128 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
0answers
36 views

Prove divergence of a improper integral

I have the improper integral $$\int ^{ +\infty }_{0}\dfrac {\sin \left( x^{q}\right) }{x^{p}}dx$$I have concluded that when $$(p-q-1)(p+q-1)\geq0$$The improper diverges, but how can I prove it?
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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
76 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
20 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
506 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
86 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
3answers
147 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
45 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) ...