0
votes
2answers
34 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
107 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
179 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
75 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
256 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
136 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
264 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
524 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
23 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
22 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
53 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
102 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
24 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
62 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
111 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
35 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
66 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
172 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
66 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
66 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
605 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
30 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
106 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
262 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 ...
1
vote
2answers
54 views

a question about integral with parameter variables?

I have a problem proving $$\int_{0}^\infty dx {\left(\int_{0}^\infty e^{-x^2t}\sin t\, dt\right)}=\int_{0}^\infty dt\left( \int_{0}^\infty e^{-x^2t}\sin t\, dx\right)$$. I have been struggling for ...
4
votes
1answer
43 views

Does the integral $\int_0^1\frac{\sin x}{\sqrt{x^3}}\cos\left(\frac1x\right)dx$ converge?

I tried to prove convergence this way: Suppose: $f(x)=\left|\frac{\sin x\cos\left(\frac1x\right)}{\sqrt{x^3}}\right|$, $g(x)=\left|\frac{\cos\frac1x}{\sqrt x}\right|$. $\lim_{x\to ...
0
votes
1answer
63 views

a question about integral proof: $\lim_{n\to \infty} \int_{0}^\infty {n\cdot {\ln(1+{f(x)\over n}}})dx=\int_{0}^\infty f(x)dx$

A non-negative function ${\rm f}\left(x\right)$ is continuous in $(0,\infty)$ and $\displaystyle{\int_{0}^{\infty}{\rm f}\left(x\right)\,{\rm d}x}$ is convergent. Then, we need to prove $$\lim_{n\to ...
0
votes
0answers
21 views

Estimates of an integral

I came across the following type of integral $$ I(r)=\int_r^{\infty}\frac{(\sinh s)^{1-\frac{n}{2}}}{\sqrt{\cosh s-\sinh r}}e^{is} ds $$ Can I have the bound $I(r)\leq C r^{-\frac{n-1}{2}}$ as $r\to0 ...
0
votes
1answer
23 views

A question about convergence of improper parametric integral

Could you give me some hint how to find all $\alpha\in R$ for with the integral $\int_0^1 \frac{a-x^{\alpha}}{1-x}$ converges. Is clear that this integral converges for all$\alpha\in N$, but I could ...
2
votes
2answers
98 views

Integration problem: $\int _ {-\infty} ^ {\infty} \frac {e^{-x^2}}{\sqrt{\pi}} e^x\ dx$

I have to integrate $$\int _ {-\infty} ^ {\infty} \frac {e^{\large-x^2}}{\sqrt{\pi}} e^{\large x}\ dx.$$ I've already done by numerical approximations, like Simpson's rule and Gauss-Hermite, but I ...
1
vote
2answers
39 views

Prove that $\int_{0}^{\infty}\left|e^{-xt}\frac{1-\cos(x)}{x}\right|dx$ converges

As part of one calculation I want to show that the following integral converges absolutely: $$\int_{0}^{\infty}e^{-x}\frac{1-\cos(x)}{x}dx$$ wihtout calculating its value. Using integral handbooks ...
3
votes
2answers
64 views

How to prove divergence of the integral $\int_{0}^{\infty}\frac{\sin(x)}{x^{2}}dx$

I want to show that the following integral diverges: $$\int_{0}^{\infty}\frac{\sin(x)}{x^{2}}dx$$ I used the substitution $ t = \frac{1}{x} $ to transform this integral into $$\int_{0}^{\infty}\sin ...
1
vote
1answer
68 views

Show that the improper integral $\int_1^\infty f(x) \ dx$ exists iff $\sum_1^\infty a_n$ converges.

The assignment: Let $(a_n)_{n\in\mathbb{N}}$ be a sequence of real numbers and $f: [1, \infty) \rightarrow \mathbb{R}$ be a function, defined by $f(x) = a_n$, for $x \in [n,n+1).$ Show that: ...
3
votes
1answer
54 views

When $\int_{0}^{\infty}f(x)dx=\sum_{n=0}^{\infty}\int_{n}^{n+1}f(x)dx$?

Is the following always true? (i.e. if both converges, limits are equal; if one diverges, the other must diverge; EXCLUDE the case where the limit keeps "jumping") $$ ...
2
votes
0answers
33 views

Question about convergence of improper integral

Could you give me some hint how to solve this problem: Suppose $f$ is continuous on $(0,1]$ and there is $M$ such as $\left|\int_x^1f(t)\, dt \right|\le M$. Prove that $\int_0^1f(x)\, dx$ converges ...
1
vote
2answers
54 views

Compute an improper integral.

Suppose $A = [0,\infty) \times [0, \infty) $. Let $f(x,y) = (x+y)e^{-x-y} $. How can I find $ \int_A f $? I know since $f$ is continuous on $A$, then $\int _A f $ exists, Do I need to evaluate $$ ...
5
votes
5answers
207 views

Evaluating the integral $\int_0^\infty \frac{x \sin rx }{a^2+x^2} dx$ using only real analysis

Calculate the integral$$ \int_0^\infty \frac{x \sin rx }{a^2+x^2} dx=\frac{1}{2}\int_{-\infty}^\infty \frac{x \sin rx }{a^2+x^2} dx,\quad a,r \in \mathbb{R}. $$ Edit: I was able to solve the integral ...
1
vote
0answers
56 views

How to prove convergence of $\int_0^1f\left(\sqrt x \right)dx$?

Could you please give me some hint how to prove convergence of $\int_0^1f\left(\sqrt x \right)dx$ when f(x) is continuous for $0<x\le1$ and $\lim_{x\to0^+}x^3f^2(x)=1$ ? I tried the usual way: ...
2
votes
1answer
91 views

How to integrate $\displaystyle\int_o^\pi\frac{dx}{\sqrt{3-\cos(x)}}$?

How to integrate $\displaystyle\int_o^\pi\frac{dx}{\sqrt{3-\cos(x)}}$ ? If I take $y=\sin\left(\frac{x}{2}\right)$ then, $\displaystyle ...
0
votes
1answer
47 views

A question about improper integral

Could you please give me some hint how to solve this problem: Suppose f(x) continuous in $[0,\infty)$ and for each a,b>0 and c>b $ab \left|\int_0^1 f\left(ax+c \right) dx \right|<1$. Prove ...
5
votes
1answer
78 views

Show that; $\displaystyle\int_0^\infty\frac{f(bx)-f(ax)}{x}dx=(l-f(0))\ln\frac{b}{a}$

Assume that $f\in C^1([0,\infty))$, $f'$ is monotonic on $[0,\infty)$ and $\lim\limits_{x\to\infty} f(x)=l$ is finite. If $a,b>0$, prove that; ...