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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1
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0answers
22 views

Can I do the following when solving my integration??

I appreciate any feedback for my question. I have an integration as follows $$\int_{-\pi}^{\pi}\frac{1}{2\pi} \prod_i \frac{1}{1+ x_ig(\theta)} d\theta $$ I have that $g(\theta)$ is the defined as ...
9
votes
2answers
99 views

Test for convergence $\int_0^{\infty} \frac{\sin(x)}{x+\log(x)} \ dx$

What is the easiest way to test the convergence of $$\int_0^{\infty} \frac{\sin(x)}{x+\log(x)} \ dx$$ Is it possible to only use the high school tools for that?
-1
votes
5answers
57 views

Does $\int_0^{\infty}\frac{x\hspace{1mm}dx}{x^3+1}$ converge? [on hold]

Does $\int_0^{\infty}\dfrac{x\hspace{1mm}dx}{x^3+1}$ converge? Can some explain how to approach this problem? All ideas are appreciated
0
votes
2answers
29 views

Integrating the gamma function

I assumed that $$\Gamma\left(k+\frac{1}{2}\right)=2\int^\infty_0 e^{-x^2}x^{2k}\,dx=\frac{\sqrt{\pi}(2k)!}{4^k k!} \,,\space k>-\frac{1}{2}$$ and that ...
1
vote
0answers
13 views

principal value integral with a singularity at t=1

is there any method to computge the principal value of the integral $$ P.V \int_{1}^{\infty} \frac{dt}{t^{a}(1-t)} $$ here 'a' is a positive real number
2
votes
2answers
129 views

How to evaluate $\int_0^1\frac{\tanh ^{-1}(x)\log(x)}{(1-x) x (x+1)} \operatorname d \!x$?

How to evaluate the following integral $$\int_0^1\frac{\tanh ^{-1}(x)\log(x)}{(1-x) x (x+1)} \operatorname d \!x $$ The numerical result is $= -1.38104$ and when I look at it, I have no idea how to ...
1
vote
0answers
26 views

Definite integral substitution question.

Let's say you have $$\int^\infty_0 f(x) dx$$ and you substitute $x=u^2$ so that the integral becomes, $$\int^\infty_0 f(u) du$$ or $$\int^{-\infty}_0 f(u) du$$ My question is, which of these ...
1
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0answers
26 views

Does there exist such function?

Fix an integer value $k\geq 1$. Let $[0,1]$ the unit interval and let $s\in [0,1]$. Does there exist a function $f$ (which depends on $k$ of course but not on $s$) such that $$\int_s^1 \left( ...
2
votes
1answer
21 views

Improper integral: is it convergent?

Is this integral finite? $$\int_s^t \frac{dx}{x^{1/2} - s^{1/2}}$$ where $s,t \in (0,\infty)$. More generally, I have the following integral $$\int_s^t ...
5
votes
5answers
98 views

Find $\int_{ - \infty }^{ + \infty } {\frac{1} {1 + {x^4}}} \;{\mathrm{d}}x$

How can we find the integral: $$\int_{ - \infty }^{ + \infty } {\frac{1} {1 + {x^4}}} \;{\mathrm{d}}x$$ I tried to find and got it to be $\cfrac{\pi}{\sqrt2}$. Am I correct? Please help me with an ...
-1
votes
0answers
22 views

Dirac's delta, unit step function integration [on hold]

Where $\delta(t)$ is dirac's delta, and $\tau$ is just a variable of integration.
7
votes
3answers
81 views

Improper integral : $\int_0^{+\infty}\frac{x\sin x}{x^2+1}$ [on hold]

How to evaluate the following improper integral : $$\int_0^{+\infty}\frac{x\sin x}{x^2+1}\,dx$$ I have tried integration by parts and variable substitution, but it didn't work.
4
votes
4answers
177 views

Inverse Trigonometric Integrals

How to calculate the value of the integrals $$\int_0^1\left(\frac{\arctan x}{x}\right)^2\,dx,$$ $$\int_0^1\left(\frac{\arctan x}{x}\right)^3\,dx $$ and $$\int_0^1\frac{\arctan^2 x\ln x}{x}\,dx?$$
3
votes
3answers
84 views

Evaluate integral: $\int_0^{+\infty}\frac{\cos{bx}-\cos{ax}}{x}dx$

It seems that $\displaystyle\int_0^{+\infty}\frac{\cos x}{x}$ is divergent, so how to solve this problem? $$\int_0^\infty \frac{\cos bx -\cos ax}{x}\, dx\quad,\quad\mbox{where}\,a,b>0$$ It's ...
5
votes
2answers
158 views

Evaluating $\int_0^1 \frac{t^{a-1}}{1-t}-\frac{ct^{b-1}}{1-t^c}\ dt$

At first sight it looks like the integral below $$\int_0^1 \frac{t^{a-1}}{1-t}-\frac{ct^{b-1}}{1-t^c}\ dt$$ can be evaluated by using some geometric series. What else can we do? Is there a fast easy ...
0
votes
0answers
37 views

Is this integral in its most simplified form?

The following integration $$F(x)= \int_{x}^{+\infty} \frac{t}{1+t^\alpha} dt$$ cannot be solved in general, however can be expressed when $\alpha=4$ as $$F(x)= 0.5 \text{tan}^{-1} (x^{-2}) $$ it can ...
5
votes
3answers
127 views

Some integral representations of the Euler–Mascheroni constant

What kind of substitution should I use to obtain the following integrals? $$\begin{align} \int_0^1 \ln \ln \left(\frac{1}{x}\right)\,dx &=\int_0^\infty e^{-x} \ln x\,dx\tag1\\ &=\int_0^\infty ...
2
votes
1answer
24 views

Find the Values of $p$ and $q$ such that $\int_0^{\infty} x^pln(1+x)^q dx$ Converges or Diverges.

The question is to find the value values of $p$ and $q$ such that the improper integral converges or diverges. My friend indeed found some values of $p$ and $q$ such that it converges, but after many ...
2
votes
2answers
31 views

How to prove this multivariable integral identity?

By numerical experimentation I found that $$ \lim_{\beta \rightarrow \infty} \frac 1 \beta \int_0^{\beta}dx \int_0^{\beta}dy \, f\left( |x-y| \right) = 2\int_0^{\infty}dx \, f(x) $$ if $f:\mathbb{R} ...
25
votes
1answer
1k views

The Wicked Integral

My brother's friend gave me the following wicked integral with a beautiful result \begin{equation} {\Large\int_0^\infty} \frac{dx}{\sqrt{x} \bigg[x^2+\left(1+2\sqrt{2}\right)x+1\bigg] ...
2
votes
1answer
27 views

Integral test error approximation

Find an N so that: $\sum_{n=1}^\infty \frac{1}{n^4}$ is between $\sum_{n=1}^N \frac{1}{n^4}$ and $\sum_{n=1}^N \frac{1}{n^4} + 0.005$ I am getting N = 200. Is this correct? I don't know if I am ...
0
votes
1answer
14 views

inverse fourier transform of exponencial

Show that $F^{-1}(e^{-|x|}) =(\sqrt{2}/\sqrt{\pi})*1/(1+x^2)$ on $\mathbb R$. $F^{-1}$ is the inverse Fourier transform. Any help? how do you solve the integrals?
0
votes
1answer
16 views

Cauchy Principal Value different from the improper integral

I am having trouble formulating an example for which $\mathcal{P}\int^{\infty}_{-\infty}f(x)dx\neq\int^{\infty}_{-\infty}f(x)dx$ Would an example be $f(x)=1/x$ because of the asymptote at $x=0$?
2
votes
2answers
50 views

Prove that $\int_{0}^{+\infty} u^{s-1} \cos (a u) \:e^{-b u}\:du=\frac{\Gamma(s)\cos\left(s\arctan \left(\frac{a}{b}\right)\right)}{(a^2+b^2)^{s/2}}$

From the answer of this OP: Ramanujan log-trigonometric integrals, I found the following formula $$\begin{align} & \int_{0}^{+\infty} u^{s-1} \cos (a u) \:e^{-b u}\:\mathrm{d}u = \Gamma ...
3
votes
1answer
46 views

Why do we care about the 'rapidness' for convergence?

It is those puzzeling improper integrals that I can't get my head around.... Does the (improper) integral $\frac 1{x^2}$ from 1 to $\infty$ coverges because it is converging "fast" or because it has ...
1
vote
0answers
20 views

Determine whether $\lim_{R\to\infty}\int_0^R\frac{|\sin x|}{x}dx-\frac{2}{\pi}\ln R$ exists

Let $$J(R):=\int_0^R\frac{|\sin x|}{x}dx.$$ (i) Show that $$\lim_{R\to\infty}\frac{J(R)}{\ln R}$$ exists and determine its value (ii)Does $$\lim_{R\to\infty}J(R)-\frac{2}{\pi}\ln R$$ exist? If ...
6
votes
3answers
152 views

How to evaluate $\int_0^\infty \frac{1}{x^n+1} dx$ [duplicate]

Noticed that the integral $$\int_0^\infty \frac{1}{x^n+1} dx$$ is often approached with partial fraction decomposition, but this gets increasingly ugly as $n$ gets bigger. Is there a neat trick to do ...
0
votes
1answer
15 views

Tempered distribution and primitive integral

$f$ is a Schwartz function on $\mathbb{R}$. Define $g(x)= \int_{-\infty}^{x} f(x)dx$. Show that $g(x)$ is a tempered distribution. Any ideas? I have no idea how to do the problem
1
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0answers
16 views

Is it always true that an integral diverges on (a,b) if it diverges on (c,b) and a<c<b?

I know that the integral of 1/x diverges on (0,b). I know the same integral still diverges on (-b, b), despite the fact it seems it should be zero. My question then is, "Is it ALWAYS the case that ...
4
votes
1answer
65 views

Evaluating the integral $ \int_0^{\infty} \cos(x^2)\, \mathrm{d} x$?

Is it necessary to make use of the Gaussian integral and the complex exponential form of the cosine in evaluating the following integral? $$\int_0^{\infty} \cos(x^2)\, \mathrm{d} x$$ Just curious - ...
1
vote
1answer
55 views

Evaluate the integral if possible

Evaluate the following integral, if possible: $$\int_1^4 \frac{w}{w-3}dw$$ $$u= w-3,\; du = dw$$ $$\int_1^4 \frac{u+3}{u}du \Rightarrow \int_1^4 \frac{u}{u}du + \int_1^4 \frac{3}{u}du$$ ...
2
votes
0answers
50 views

difficult integral [duplicate]

Evaluate the following integral : $\int _0^{\infty }\:\left(\frac{^{\:}\sin \left(x\right)}{x}\right)^3dx$
1
vote
2answers
54 views

Evaluate the integral $\int_{1}^{4} w/(w-3) dw$ if possible

Evaluate the following integral, if possible: $$\int_{1}^{4} \frac{w}{w-3} dw.$$ I am supposed to be using improper integrals so I know I should find $$ \lim_{t \rightarrow3^-} \int_{1}^{t} ...
10
votes
1answer
166 views

Two integral involving logarithm and polylogarithm function

Evaluate the following integrals $$\int_0^1\frac{\ln(1-x)}{x}\text{Li}_3\left(\frac{1 + x}{2} \right)\,dx\\ .\\ \int_0^1\frac{\ln^2(1-x)}{x}\text{Li}_2\left(\frac{1 + x}{2} \right)\,dx$$
1
vote
1answer
36 views

Check the convergence (& absolutely) of parametric integral [closed]

$$\int\limits_{-1}^{1} \left(\frac{1+x}{1-x}\right)^{\alpha} ln(2+x)dx$$ Don't know where to start..
1
vote
1answer
38 views

Calculating improper integral

Does anyone know how to solve the following integral: $$I =\int_{0}^\infty \cos(t \mathrm{log}( x))\,\mathrm{e}^{-ax}\, \mathrm{d}x,$$ where $t$ and $a$ are real. Please show some intermediate ...
5
votes
2answers
95 views

About Integration

How to calculate the following integral $$ \int \frac{\tanh(\sqrt{1+z^2})}{\sqrt{1+z^2}}dz $$ Is there any ways to calculate those integral in analytic? (Is $[0,\infty]$, case the integral is ...
1
vote
0answers
55 views

Proof of Definite Integral of Even Function for Improper Integrals

I am trying to prove $\displaystyle \int_{\mathop \to -a}^{\mathop \to a} f \left({x}\right) \ \mathrm d x = 2 \int_0^{\mathop \to a} f \left({x}\right) \ \mathrm d x$ for $f$ which is an even ...
2
votes
1answer
63 views

Calculating a complex definite improper integral: $I= \int_{0}^\infty x^{it}\,\mathrm{e}^{-ax}\, dx$

Does anyone know how to find the value of this integral: $$I= \int_{0}^\infty x^{it}\,\mathrm{e}^{-ax}\, dx,$$ where $i=\sqrt{-1}$ and $t$, $a$ are real. Please give me a hint. Thank you.
10
votes
3answers
142 views

Evaluate $\int_{0}^{\infty} \mathrm{e}^{-x^2-x^{-2}}\, dx$

I have to find $$I=\int_{0}^{\infty} \mathrm{e}^{-x^2-x^{-2}}\, dx $$ I think we could use $$\int_{0}^{\infty} \mathrm{e}^{-x^2}\, dx = \frac{\sqrt \pi}{2} $$ But I don't know how. Thanks.
7
votes
3answers
93 views

Calculate trigonometric integral $ \int_{-\infty}^{\infty}{\sin(x^2)}\,dx$

Recently, I came across the following integral: $$ \int_{-\infty}^{\infty}{\sin(x^2)}\,dx=\int_{-\infty}^{\infty}{\cos(x^2)}\,dx=\sqrt{\frac{\pi}{2}} $$ What are the different ways to calculate such ...
4
votes
2answers
70 views

How to compute $\int_0^{\infty} x^{t-1} e^{-x}\ln(x)\,dx$?

I have hit the following integral (in the process of trying to derive a finite-sample correction for the Maximum Likelihood fitting of the Generalized Extreme Value distribution...): ...
6
votes
3answers
153 views

Evaluate $\int^1_0 \log^2(1-x) \log^2(x) \, dx$

I have no idea where to even start. WolframAlpha cant compute it either. $$\int^1_0 \log^2(1-x) \log^2(x) \, dx$$ I think it can be done with series, but I am not sure, can someone help a little so ...
1
vote
0answers
12 views

How naturally can functions defined by parametric integrals be interpolated from $\mathbb N$ to $\mathbb R^+$?

This question has not got any answer on Mathoverflow. I admit that it is unusual to cross-post in this direction (from MO to math.SE), but knowing that some of those really unbelievable integrals tend ...
1
vote
3answers
35 views

Integral convergence and limit question

I have a question that has come up in a group homework project that neither I nor my partner have any idea how to solve. I'm hoping someone can give me a hint or some guidance as to how to go about ...
0
votes
0answers
18 views

convergence of sequence of functions with finite second moment

Given $0<a<1$. Let $\phi:\mathbb R\mapsto\mathbb R$ is defined by $\phi(x)=\frac1{\sqrt{2\pi}}e^{-\frac{x^2}2}$ for all $x\in\mathbb R$. Suppose we are given a sequence of functions $\{f_n\}$ ...
0
votes
1answer
27 views

Convergence uniformly implies in integral

Let $\phi:\mathbb R\mapsto\mathbb R$ is defined by $\phi(x)=\frac1{\sqrt{2\pi}}e^{-\frac{x^2}2}$ for all $x\in\mathbb R$. Suppose we are given a sequence of functions $\{f_n\}$ such that ...
3
votes
4answers
48 views

What do limits of functions of the form $te^t$ have to do with l'Hopital's rule?

I have an improper function that I have to integrate from some number to infinity. Once integration is done, the function is of the form $te^t$. What I'm wondering is what does this have to do with ...
1
vote
1answer
27 views

Solve $\int\limits_{-\infty}^{\infty}e^{-cx^2}\sin(sx)dx $

How to prove that $$\int\limits_{-\infty}^{\infty}e^{-cx^2}\sin(sx)dx = 0,$$ where $c>0$?
1
vote
1answer
20 views

Integral solution using modified Bessel function of order 1

How to solve this integral making use of Modified Bessel function of order one? $\frac{q\gamma b}{2\pi}\int_{-\infty}^{\infty}(\gamma ^{2}\nu ^{2}t^{2}+b^{2})^{-3/2}\exp (i\omega t)dt$ Context: I ...