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.

learn more… | top users | synonyms

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
20 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
40 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
56 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?
7
votes
0answers
74 views

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
39 views

what is the result of an infinity minus pi [on hold]

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
109 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
35 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
95 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}$ [on hold]

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
68 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
79 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 ...
2
votes
1answer
69 views
+50

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? $$\mathcal{I}= \int_0^\infty \frac{x^{23}}{(5x^2+7^2)^{17}}\,\mathrm{d}x$$
0
votes
1answer
24 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: $I(t) = \displaystyle\int_{t}^{1}x^{a}\mathrm{d\alpha(x)}$ ...
-1
votes
0answers
44 views

Closer form for $I=\int_0^{\pi/2}\frac{x^2\log{(\cos{x})}}{\sin^2x}\,dx$ [closed]

Closer form for $$I=\int_0^{\pi/2}\frac{x^2\log{(\cos{x})}}{\sin^2x}\,dx$$ The nunerical value is -2,046858918488525363347483... At first sight, this integral can not be expressed by means of ...
0
votes
3answers
80 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
15 views

Numerical Triple integral with three other parameters in R

I am trying to integrate this function $f(u,v,w; t,x_{0},z)$ with respect to three variables, $u$, $v$, $w$, although the function also have other three parameters $t$, $x_0$, and $z$. Question: How ...
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 ...
-1
votes
1answer
89 views

Please help me with this definite integral problem [closed]

$$\int_2^3\frac{x\,\textrm{d}x}{\sqrt{x(x-1)(x-2)(x-3)(x-4)}}$$
7
votes
4answers
127 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
58 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
35 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?
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
75 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
18 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
504 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
55 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
145 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) ...
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} - ...
3
votes
4answers
305 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
49 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
368 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
30 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 ...
-2
votes
1answer
40 views

Direct comparison test for ( Improper ) Integrals [closed]

How we can prove with direct comparison test for ( Improper ) Integrals that is bounded: $\int _1^n\:e^{-x^3}dx$ ?
1
vote
0answers
39 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 ...