1
vote
4answers
45 views

How to prove that $\int_{0}^{\infty}{\frac{e^{-nx}}{\sqrt{x}}}dx$ exists

I am trying to show that the integral $\int_{0}^{\infty}{\frac{e^{-nx}}{\sqrt{x}}}dx$ exists ($n$ is a natural number). I tried to use the comparison theorem by bounding from above the integrand by ...
1
vote
0answers
34 views

Determining the sets of alpha for which some (Riemann, Lebesgue - integrals) exists

$$\int_0^{\infty} \frac{\sin(x)}{x^{\alpha}} \, dx.$$ $$\int_{[0, \infty]} \frac{\sin(x)}{x^{\alpha}} \, d \lambda(x).$$ $$\int_{\Bbb R^2} \frac{\sin(\| x \|)}{\| x \|^{\alpha}} \, d \lambda_2 ...
13
votes
2answers
364 views
+100

Closed form for the integral $\int_{0}^{\infty}\frac{\ln^{2}(x)\ln(1+x)}{(1-x)(x^{2}+1)}dx$

Here is a challenging one maybe some would like a go at. Show that: ...
12
votes
5answers
287 views

The other ways to calculate $\int_0^1\frac{\ln(1-x^2)}{x}dx$

Prove that $$\int_0^1\frac{\ln(1-x^2)}{x}dx=-\frac{\pi^2}{12}$$ without using series expansion. An easy way to calculate the above integral is using series expansion. Here is an example ...
9
votes
3answers
116 views

Proof of $\int_0^\infty \frac{x^{\alpha}dx}{1+2x\cos\beta +x^{2}}=\frac{\pi\sin (\alpha\beta)}{\sin (\alpha\pi)\sin \beta }$

I found a nice formula of the following integral here $$\int_0^\infty \frac{x^{\alpha}dx}{1+2x\cos\beta +x^{2}}=\frac{\pi\sin (\alpha\beta)}{\sin (\alpha\pi)\sin \beta }$$ It states there that ...
9
votes
5answers
240 views

Prove that $\int_0^1\frac{1-x}{1-x^6}\ln^4x\,dx=\frac{16\sqrt{3}}{729}\pi^5+\frac{605}{54}\zeta(5)$

This integral comes from a well-known site (I am sorry, the site is classified due to regarding the OP.) $$\int_0^1\frac{1-x}{1-x^6}\ln^4x\,dx$$ I can calculate the integral using the help of ...
10
votes
3answers
191 views

Prove that $\int_0^1\frac{\ln(1-x)\ln^2x}{x-1}dx=\frac{\pi^4}{180}$

Prove that (please) $$\int_0^1\frac{\ln(1-x)\ln^2x}{x-1}dx=\frac{\pi^4}{180}$$ I've tried using Taylor series and I ended up with $$-\sum_{m=0}^\infty\sum_{n=1}^\infty\frac{2}{n(m+n+1)^3}$$ I am ...
1
vote
1answer
37 views

Improper integral $\int_{B}\frac {1}{|x|^\alpha}dV$

Let B be the ball $|x|\le 1$, $x\in R^n$. For what $\alpha$ does $$\int_{B}\frac {1}{|x|^\alpha}dV$$ exists? I find it hard when it comes to generalize this statement in $R^n$. I've been able to do ...
4
votes
0answers
52 views

Improper multivariable integrals

I'm having trouble with the integral $$\iiint_{1\le x^2+y^2+z^2 }\frac{\mathrm{d}x~\mathrm{d}y~\mathrm{d}z}{xyz}$$ this is what I've done so far: $$\lim_{b\to +\infty}\int_1^b \int_0^{2\pi} ...
0
votes
0answers
27 views

Uniformly continuous function which is integrable but does not have a limit [duplicate]

Is there an example of a function $f:[0,+\infty)\to \mathbb{R}$ which is uniformly continuous and $\int_0^{+\infty}|f(x)|dx<+\infty$, but $\lim_{x\to+\infty}f(x)\neq0$ (since it is integrable this ...
1
vote
1answer
38 views

Convergence of the improper integral $\int_{0}^{\infty}\frac{x^{p-1}}{1+qx}dx$

I found that the following converges where ${0<p<1}$,and ${0<q}$, but I'm having some trouble where q is negative. Because it has some "blow up" point, it seems to diverge, but i'm not ...
4
votes
3answers
270 views

Prove $\int_{\mathbb{R^{+}}} \frac{\sin^3 {(\pi x^2)} \cos {(4x^2)}}{x^5} dx=\frac{\pi}{32} (3\pi-4)^2$

How do you arrive at the result $$I=\displaystyle\int_{\mathbb{R^{+}}} \dfrac{\sin^3 {(\pi x^2)} \cos {(4x^2)}}{x^5} dx=\dfrac{\pi}{32} (3\pi-4)^2\ ?$$ Wolfram Alpha agrees numerically. I tried ...
1
vote
3answers
133 views

Value of the integral $\int_{\mathbb{R}} \frac{x\sin {(\pi x)}}{(1+x^2)^2}$

How do we evaluate the integral $$I=\displaystyle\int_{\mathbb{R}} \dfrac{x\sin {(\pi x)}}{(1+x^2)^2}$$ I have wasted so much time on this integral, tried many substitutions $(x^2=t, \ \pi x^2=t)$. ...
9
votes
2answers
171 views

An exercise from my brother: $\int_{-1}^1\frac{\ln (2x-1)}{\sqrt[\large 6]{x(1-x)(1-2x)^4}}\,dx$

My brother asked me to calculate the following integral before we had dinner and I have been working to calculate it since then ($\pm\, 4$ hours). He said, it has a beautiful closed form but I doubt ...
2
votes
2answers
60 views

Duo Fresnel-like integrals $(??)$

I really wonder how I can prove the following integrals. $$\int_0^\infty \sin ax^2\cos 2bx\, dx=\frac{1}{2}\sqrt{\frac{\pi}{2a}}\left(\cos \frac{b^2}{a}-\sin\frac{b^2}{a}\right)$$ and ...
4
votes
2answers
87 views

Evaluate $\int_{-\infty}^{\infty}\int_{-\infty}^{\infty}e^{-\frac{1}{2}(x^2-xy+y^2)}dx\, dy$

I need to evaluate the following integral: $$\int_{-\infty}^{\infty}\int_{-\infty}^{\infty}e^{-\frac{1}{2}(x^2-xy+y^2)}dx\, dy$$ I thought of evaluating the iterated integral ...
9
votes
1answer
179 views

Evaluate $\int_{0}^{\large\frac{\pi}{4}} \ln {(\sin x)}\cdot\ln {(\cos x)} \left(\frac{\ln{(\sin x)}}{\cot x}+\frac{\ln {(\cos x)}}{\tan x}\right)dx$

How do I find the value of this integral? $$I=\int_{0}^{\Large\frac{\pi}{4}} \ln {(\sin x)}\cdot\ln {(\cos x)} \left(\dfrac{\ln{(\sin x)}}{\cot x}+\dfrac{\ln {(\cos x)}}{\tan x}\right)dx$$ I tried ...
0
votes
2answers
41 views

Convergence of improper integral of $\ln f(x)$

Is there something know about the convergence of $\int_0^1 \ln f(x)dx $ for $f(x)$ continous on $\left(0,1\right)$ and both limits exists, i.e. $\lim_{x\to 0} f(x)$ and $\lim_{x\to 1} f(x)$ ? I ...
5
votes
3answers
117 views

How to $\int_{0}^\infty {\sin^3(x)\over x}dx$

How to evaluate : $$\int_{0}^\infty {\sin^3(x)\over x}dx$$ I don't know how to do it. I tried to finish it using integration by parts, but it doesn't work? Can someone tell me how to evaluate the ...
6
votes
5answers
209 views

An improper integral : $\int_{0}^\infty {\ln(a^2+x^2)\over{b^2+x^2}}dx$

How to evaluate the following improper integral:$$\int_{0}^\infty {\ln(a^2+x^2)\over{b^2+x^2}}dx,$$ where $a,b>0$. I tried to suppose $$f(a)=\int_0^\infty {\ln(a^2+x^2)\over{b^2+x^2}}dx,$$ based ...
1
vote
0answers
52 views

Question on integral

I need a confirmation and answer about the following problem: If we have $ g(x)=\ln x+{ x }^{ -1/2 }{ 1 }_{ x\le 1 } $ I'm trying to determine $ \int _{ 0 }^{ +\infty }{ \ln x+{ x }^{ -1/2 }{ 1 }_{ ...
4
votes
1answer
78 views

a complicated question about double improper integral

how to evaluate $$\iint_{y\ge x^2+1}{dx\,dy\over{x^4+y^2}}$$ My solution: the initial intergral $$ =2\int_0^\infty \left(\int_{x^2+1}^\infty {dy\over {x^4+y^2}}\right)\,dx = \int_0^\infty ...
8
votes
1answer
260 views

Evaluation of $\int_0^1 \frac{\log(1+x)}{1+x}\log\left(\log\left(\frac{1}{x}\right)\right) \ dx$

I need some hints, clues for getting the closed form of $$\int_0^1 \frac{\log(1+x)}{1+x}\log\left(\log\left(\frac{1}{x}\right)\right) \ dx$$
2
votes
4answers
137 views

Prove that $\lim_{n\rightarrow \infty} \int_{0 }^{\pi} \frac{\sin(nx)}{nx}dx=0$

Prove that $$\underset{n\rightarrow \infty }{\lim} \ \int_{\epsilon }^{\pi} \frac{\sin(nx)}{nx}dx=0\ ;\ \epsilon>0$$ then use the result to deduce: $$\underset{n\rightarrow \infty }{\lim} \ ...
4
votes
3answers
272 views

Integral without residues

How do I do this integral without using complex variable theorems? (i.e. residues) $$\lim_{n\to \infty} \int_0^{\infty} \frac{\cos(nx)}{1+x^2} \, dx$$
0
votes
1answer
35 views

Convergence of improper integral depending on parameter [closed]

For what $a>0$ $(\frac{1}{\sin x})^a$ is integrable on $(0,\frac{\pi}{2})$?
4
votes
3answers
526 views

Prove the equation

Prove that $$\int_0^{\infty}\exp\left(-\left(x^2+\dfrac{a^2}{x^2}\right)\right)\text{d}x=\frac{e^{-2a}\sqrt{\pi}}{2}$$ Assume that the equation is true for $a=0.$
2
votes
1answer
23 views

Using Taylor's series in imporper integrals

Is it possible to simplify an improper integral using Taylor's series? How can I prove this procedure is correct? For example, take $$f(\alpha)=\int_0^{\infty} ...
1
vote
0answers
24 views

Existence of measurable fuction on non-atomic measure space whose integral is infinity

Let $(X,M,\mu)$ be non atomic measure space with $\mu(X)>0.$ Show that there is a measurable function $f:X\to [0,\infty),$ for which $\int f(x)d\mu(x)=\infty.$ No idea at all. I am preparing for ...
1
vote
1answer
25 views

Convergence of an Improper Integral $\int_{-\infty}^{\infty}\cos(x\log\left|x\right|)dx$

This is a question from an old exam qualifier: Show that the improper integral $\int_{-\infty}^{\infty}\cos(x\log\left|x\right|)dx$ is convergent. I first notice that \begin{equation*} ...
0
votes
2answers
54 views

Finding $(p,q)$ such that $\frac{x^p}{1+x^q}$ is integrable on $(0,+\infty)$

I'm trying to show that $f(x) = \frac{x^p}{1+x^q}$ is integrable on $(0,\infty)$ if and only if $p > -1$ and $q-p > 1$. So on $[1,\infty)$ we can compare with $g(x) = x^{p-q}$ which is ...
1
vote
3answers
105 views

Evaluate $\int_0^{\infty}\int_0^{\infty}e^{-x^2-2xy-y^2}\,dx\,dy$

I would like to compute the following, $$ \int_0^{\infty}\int_0^{\infty}e^{-x^2-2xy-y^2}\ dx\,dy $$ It is obvious that we can rewrite the integral above to, $$ ...
4
votes
4answers
137 views

Question about $\frac{\sin(x)}{x}$ and $\frac{\cos(x)}{x}$

So here is my question. As known the famous integral $$ \int_0^{\infty} \frac{\sin(x)}{x}dx$$ converges an its value is $\frac{\pi}{2}$. As I was trying to solve a different integral today, after ...
0
votes
1answer
25 views

Would like to compute the limit of some integral sequence

Consider $$ \lim_{n\rightarrow \infty}\int_{\mathbb R}e^{-|x|n}e^{\frac{x^2}{2}}dx $$ The goal of an exercise I am working on is to compute the limit of the the integral above. By intuition it should ...
1
vote
2answers
49 views

Give an example of $f$ that is $C^\infty$, $\int_0^\infty f(t)dt$ converges but $f$ does not converge.

Today, my teacher asked about a real function that was integrable, but such that $\displaystyle\lim_{t\to \infty}f(t)\neq 0$ ($f$ need not have a limit). It's easy to forge an example: a function ...
0
votes
1answer
66 views

How to prove that integral of function is convergent

$\int_{0}^{\infty} \frac{(\sin(x) )}{ x} \,\mathrm dx$ and $\int_{0}^{\infty} \frac{(\sin(x) \arctan(x))}{ x} \,\mathrm dx$ These are convergent. How to prove that?? I using the comparison test. ...
2
votes
3answers
112 views

Value of $\int ^\infty_0 \frac{b\sin{ax} - a\sin{bx}}{x}dx$?

$$\int ^\infty_0 \frac{b\sin{ax} - a\sin{bx}}{x}dx$$ Hello guys! I'm having trouble solving this integral...looks an awful lot like an Frullani Integral, and I've tried to get it to an appropriate ...
0
votes
0answers
38 views

Find the value of the integral $\int^\infty_0 \frac{\cos(ax)}{1+x^2} dx$ [duplicate]

I have to compute the value of the integral $$ \int^\infty_0 \frac{\cos{(ax)}}{1+x^2} dx $$ It may help that $$\int_0^\infty e^{-tx}\cdot \sin(x)dx = \dfrac{1}{1+t^2}$$ but i can't find the link ...
3
votes
3answers
74 views

Compute $\int_0^\infty \dfrac{e^{-tx}\sin(x)}{x}dx$

I have to compute$$\int_0^\infty \dfrac{e^{-tx}\cdot \sin(x)}{x}dx$$ This is following a helping problem $$\int_0^\infty e^{-tx}\cdot \sin(x)dx$$ which using IPB two times turned out to be ...
2
votes
2answers
67 views

a question about summation of series, how to prove $\int_0^\infty e^{-x}S(x)$=$\sum_{i=0}^\infty a_nn!$

If the coefficients of $\sum_{n=0}^\infty a_nx^n$ is non-negative($a_n\ge 0$ for every n),and the sum function is S(x). Also,suppose$\sum_{i=0}^\infty a_nn!$ is convergent,please prove $\int_0^\infty ...
5
votes
5answers
88 views

a question about a complex integral, I am struggling with it!

How to prove $$\int _0^1 {\ln(x)\over{1-x^2}}={-\pi^{2}\over 8}$$ My solution: If we can prove$\int _0^1 {\ln(x)\over{1-x^2}}= \lim_{n\to \infty} \int _0^1\ln(x)(1+x^2+x^4+......+x^{2n})$,then I ...
3
votes
3answers
175 views

Integral $\int_0^{\infty} \frac{x^{a-1}}{1+x} dx $ converges?

For what values ​​of $a \in \mathbb{R}$ the following integral converges? $$\int_0^{\infty} \frac{x^{a-1}}{1+x}\ dx $$ I tried to compute the integral but I stuck solving and then I tried to compare ...
2
votes
1answer
35 views

A Subadditive Function Satisfying an Integrability Property

Is there a function $f: [0,\infty) \rightarrow [0,\infty)$ that is Subadditive $\left(f(x + y) \leq f(x) + f(y), \forall x,y \in [0,\infty)\right)$, Satisfies $\lim_{r \rightarrow \infty} f(r) = ...
2
votes
2answers
70 views

Integral involving a logarithm and a linear rational function

$$\int_{0}^{1} \frac{\log x}{x-1}dx$$ I was wondering: is it possible to evaluate this integral with real methods? Playing around with a series expansion I was able to recognize that the integral is ...
3
votes
2answers
68 views

I want to prove $k(x,t)=\frac{1}{\sqrt{4\pi t} } e^{\frac{-x^2}{4t}} $

I have this integral $$u(x,t)=\int _{-\infty}^{\infty} f(\eta)\left[\frac{1}{2\pi}\int _{-\infty}^{\infty}e^{iw(x-\eta)-w^2t}\ dw\right]\ d\eta=\int _{-\infty}^{\infty}k(x-\eta,t)f(\eta)\ d\eta$$ I ...
13
votes
3answers
675 views

Surely You're Joking, Mr. Feynman! $\int_0^\infty\frac{\sin^2x}{x^2(1+x^2)}\,dx$ [duplicate]

Prove the following \begin{equation}\int_0^\infty\frac{\sin^2x}{x^2(1+x^2)}\,dx=\frac{\pi}{4}+\frac{\pi}{4e^2}\end{equation} I would love to see how Mathematics SE users prove the integral ...
1
vote
1answer
32 views

If $\int _1^{\infty }f\left(x\right)dx$ converges absolutely then $\int _1^{\infty }\sin \left(x\right)f\left(x\right)dx$ exists

I'm in need of some assistance with a homework question (I'm doing some calculus work by myself and have gotten stuck on this question): "Prove or give a counter-example of of the following ...
3
votes
3answers
107 views

Does $\int _1^{\infty }\left(\sin \left(x^2\right)\right)dx$ converge or diverge? [duplicate]

I'm in need of some assistance regarding a question in my Calculus textboox: Find if the following converges or diverges without calculating the integral: $$\int _1^{\infty }\left(\sin ...
1
vote
3answers
48 views

Let $(f_n)_{n\in\mathbb{N}} \rightarrow f$ on $[0,\infty)$. True or false: $\lim_{n\to\infty}\int_0^{\infty}f_n(x) \ dx = \int_0^{\infty}f(x) \ dx.$

The Assignment: Let $(f_n)_{n\in\mathbb{N}}$ converge uniformly to $f$ on $[0,\infty)$ and let the improper integrals of $f_n$ and $f$ exist on $[0,\infty)$. True or false: ...
5
votes
4answers
266 views

The closed form of $\int_0^{\infty} \frac{\log(\cosh(x))}{x} e^{-x} \ dx$

An integral I discussed last days in a chat, and it looks like a hard nut since after some manipulations of the initial form we reach an integral where the integrand is expressed in terms of ...