Tagged Questions

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
vote
0answers
3 views

Evaluating by real methods $\int_0^{\pi/2} \frac{x^5}{2-\cos^2(x)}\ dx$

I'm sure you guys can briefly get the result by some methods of complex analysis, but now I'm only interested in real analysis methods of proving the result. What would you propose for that? ...
2
votes
0answers
45 views

A double integration with an embedded integral which is hard to solve

I have written this in way to make it as much as possible non-confusing. I will start describing my problem and I will walk you through my question, I have a double integration which I am trying to ...
8
votes
1answer
76 views

A limit evaluating to $2 K$ (Catalan's constant)

Experimentally I discovered the limit below that says that $$\lim_{n\to\infty} \int_0^{\pi/2} \frac{1}{\displaystyle ...
20
votes
2answers
169 views

Prove the integral evaluates to $\frac{K}{\pi}$

Yesterday I received the following integral that might require some tedious steps to do $$\int_0^{\infty}{\small\left[ \frac{x}{\log^2\left(e^{\large x^2}-1\right)}- \frac{x}{\sqrt{e^{\large ...
5
votes
0answers
68 views

Finding the closed form of $\sum_{k=1}^{\infty} \sum_{n=1}^{\infty}(-1)^{k+n} \frac{\log(k+n)}{k n}$

A while ago I computed pretty easily the series $\sum_{k=1}^{\infty} \sum_{n=1}^{\infty}(-1)^{k+n} \frac{\log(k+n)}{k+ n}$ and then I thought of tackling the case where we have the product instead of ...
2
votes
0answers
58 views

A hard integral from probability theory

I am trying to resolve this integral, which comes out of considering a compound distribution of normal variables: $$ \int_{-\infty}^{\infty} \frac{1}{\sigma_{\sigma} \sqrt{2 \pi}} ...
8
votes
5answers
172 views

Improper integral of $\dfrac{x^2}{ e^x−1}$

I am trying to evaluate $$\int_0^t \frac{x^2}{ e^x−1}dx$$ for $t>0$. I tried integrating by parts, like follows: $$\int_0^t x\cdot\frac{xe^{-x}}{1-e^{-x}}dx=t\cdot Li_2(1-e^{-t})-\int_0^t ...
3
votes
1answer
81 views

Differentiation under integral

I want to compute the following integral, $\int_0^\infty\frac{dx}{(x^2 + p)^{n}}$ for any $n \in \mathbb{N}$ I've tried to add a parameter, obtaining its integral and then taking derivatives with ...
4
votes
1answer
34 views

Three integral involving polylogarithm function

$\newcommand{\Li}{\operatorname{Li}}$Evaluate the following integrals $$\int\limits_0^1 \frac{\Li_2^3(x)}{x}dx, \quad \int\limits_0^1 \frac{\Li_2^2(x)\Li_3 (x)}{x} dx, \quad \int\limits_0^1 ...
0
votes
1answer
46 views

rational exponent of negative base

I have the definite integral $$\int_{1}^{\,9} {\frac{6}{\sqrt[3]{x-9}}}\, \mathrm dx$$ When I try to evaluate it I get the indefinite integral equals $9(x-9)^{2/3}$ and evaluating at the limits gives ...
9
votes
2answers
175 views

Closed form of $\int_0^{\infty} \frac{\log(x)}{\cosh(x) \sec(x)- \tan(x)} \ dx$

What real analysis tools would you recommend me for getting the closed form of the integral below? $$\int_0^{\infty} \frac{\log(x)}{\cosh(x) \sec(x)- \tan(x)} \ dx$$
7
votes
2answers
144 views

Using differentiation under integral sign to calculate a definite integral

I want to calculate the integral $$\int^{\pi/2}_0\frac{\log(1+\sin\phi)}{\sin\phi}d\phi$$ using differentiation with respect to parameter in the integral ...
-1
votes
1answer
40 views

What is $\int_{-ia^2t}^{\infty} x^2 \exp \left ({\frac{-x^2}{2a^2}}\right ) \mathrm{d}x$?

I want to determine the integral $$\int_{-ia^2t}^\infty x^2 \exp\left ({\frac{-x^2}{2a^2}}\right ) \mathrm{d}x.$$ I guess a representation in terms of error functions should exist, but I guess that ...
0
votes
2answers
48 views

Show that $\int_0^1 |\frac{1}{x}\sin\frac{1}{x}|\ dx$ diverges

I read that the improper Riemann integral $$\int_0^1 \Bigg|\frac{1}{x}\sin\frac{1}{x}\Bigg|\ dx$$ diverges. I have tried comparison criteria for $\int_0^1 |\frac{1}{x}\sin\frac{1}{x}|dx$, but I ...
5
votes
2answers
69 views

How to evaluate $\int_0^{\infty} \bigg(\frac{e^{-x}}{\sinh(x)} - \frac{e^{-3x}}{x}\bigg) \; dx$

Evaluate the integral below $$\int_0^{\infty} \bigg(\frac{e^{-x}}{\sinh(x)} - \frac{e^{-3x}}{x}\bigg) \; dx$$ Using Wolfram I get the integral is $\gamma + \log\bigg(\frac{3}{2}\bigg)$, where ...
14
votes
1answer
136 views
1
vote
0answers
33 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 ...
7
votes
2answers
126 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
58 views

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

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
30 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
18 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
143 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
vote
0answers
28 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
108 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 [closed]

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

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

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
196 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
88 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
167 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
40 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
130 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
26 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
32 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} ...
26
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
15 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
18 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
56 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
159 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
16 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
vote
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: $$\int_1^4 \frac{w}{w-3}dw$$ $$u= w-3,\; du = dw$$ $$\int_1^4 \frac{u+3}{u}du \Rightarrow \int_1^4 \frac{u}{u}du + \int_1^4 \frac{3}{u}du$$ ...
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} ...
11
votes
1answer
187 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..