Questions tagged [improper-integrals]
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 $\infty$ or $-\infty$, or as both endpoints approach limits.
8,266 questions
4 votes
1 answer
147 views
Real variable method to show that $\int_{-\infty}^\infty \frac{\sinh ax}{\sinh \pi x}\cos bx dx = \frac{\sin a}{\cos a+\cosh b}$?
There are evaluations of the above integral using complex analytic techniques, but is there a real variables way to show it? I've tried the "Feynman differentiation under the integral trick" ...
- 140
0 votes
0 answers
22 views
Convergence Criterion of Newton Integral by Direct Comparison
I have a question regarding convergence of a Newton integral by direct comparison. That is, if we know that some function $g(x)$ is newton integrable on some interval $(a,b)$, then a function $f(x) \...
6 votes
4 answers
330 views
Is there a closed form for integral $\int_0^\infty \frac{\ln(1+x^2)}{x^2+2a x+1}dx $ with $|a|<1$
The integral converges since $|a|<1$. In the special case of $a=0$, the integral $$I(a)=\int_0^\infty \frac{\ln(1+x^2)}{x^2+2a x+1}dx $$ is well-known, which is $$\int_0^\infty \frac{\ln(1+x^2)}{1+...
- 2,958
6 votes
6 answers
349 views
Seeking other generalisations to the integral $\int_0^{\infty} \frac{\ln \left(x+\frac{1}{x}\right)}{1+x^2}dx$
The integral $$ \int_0^{\infty} \frac{\ln \left(x+\frac{1}{x}\right)}{1+x^2}=\pi \ln 2 $$ invites me to investigate the integral $$ I=\int_0^{\infty} \frac{\ln \left(x+\frac{1}{x}\right)}{x^4+1} d x $$...
- 33.2k
5 votes
2 answers
256 views
Closed form of $\int_0^{\infty} \frac{\sin (\tan x) \cos ^{2n-1} x}{x} d x?$
Being attracted by the answer in the post $$\int_0^{\infty} \frac{\sin (\tan x) }{x} d x = \frac{\pi}{2}\left(1- \frac 1e \right) , $$ I started to investigate and surprisingly found that $$ \int_0^{\...
- 33.2k
4 votes
3 answers
206 views
Other methods/ generalisations to $\int_0^{\frac{\pi}{2}} \int_0^{\frac{\pi}{2}} \frac{\tan ^{-1}(\sin x \sin y)}{\sin x} d x d y?$
Latest news Thanks a lot for the contributions from both @Luce with @FDP and @ Sangchul Lee who brought us a very decent alternative methods(checked by the Desmo) for the general integral: $$\boxed{\...
- 33.2k
1 vote
0 answers
63 views
Why does Limit Comparison Test with $g(x) = \log x$ give incomplete convergence range?
I'm trying to determine the values of $n$ for which the following integral converges: $$\int_0^1 \frac{x^n \log x}{(1+x)^2} \, dx$$ The only point of discontinuity is at $x = 0$. My attempt: I used ...
- 65
9 votes
3 answers
425 views
Generalisation of $\int_0^{\infty} \frac{\tan ^{-1}\left(\tan ^2 x\right)}{x^2}\,dx .$
Being attracted by the result $$\int_0^{\infty} \frac{\tan ^{-1}\left(\tan ^2 x\right)}{x^2}\,dx=\frac{\pi}{\sqrt 2}, $$ I tried to generalise the integral for any even integer $n\ge 2$, $$I_n=\int_0^{...
- 33.2k
15 votes
4 answers
658 views
Closed form for $I_n = \int_0^{\pi} \frac{x \sin(nx)}{1 - \cos x}\, dx$
I recently constructed the following integral for natural numbers $n$, which led me to an interesting discovery. $$I_n = \int_0^{\pi} \frac{x \sin(nx)}{1 - \cos x}dx $$ Using online tools, I obtained ...
- 1,106
4 votes
0 answers
254 views
How does one solve the integral $\int_{1}^{\infty}\frac{\ln\left(\arctan\left(x\right)\right)}{x^{2}+\ln\left(x^{2}+1\right)}\,\mathrm dx$
When I was on Twitter, I saw the integral $$\int_{1}^{\infty}\frac{\ln\left(\arctan\left(x\right)\right)}{x^{2}+\ln\left(x^{2}+1\right)}\,\mathrm dx.$$ The solution was not part of the post, but I ...
- 181
1 vote
1 answer
47 views
Stieltjes tranform of Lebesgue measure
I was trying to get a better feeling for the Stieljes transform by applying it to the Lebesgue measure restricted to $[0,1]$ but I am having an issue to finish the computation. For a Borel probability ...
- 8,566
3 votes
1 answer
137 views
How do I solve or prove the convergence of $\int_{0}^{\infty}(\sin\left(1-e^{-x}\right)-\sin\left(1\right))dx$
Does the integral $$\int_{0}^{\infty}(\sin\left(1-e^{-x}\right)-\sin\left(1\right))dx$$ converge? If so, how would I solve it? It seems to be a $\infty-\infty$ indeterminate limit but I have been ...
- 181
8 votes
1 answer
266 views
On the rigor of improper integrals
Recently I was learning to evaluate the improper integral $$ I=\int_{-\infty}^\infty\frac{du}{u^2+2} $$ My instructor said that we could write $$ I=\lim_{t\to\infty}\int_{-t}^t \frac{du}{u^2+2}=\lim_{...
- 143
0 votes
1 answer
129 views
Proof and applicability of an integral formula $\int_{0}^{\infty}\int_{0}^{\infty}\sin(xy)xf(x)\,\mathrm dx\,\mathrm dy$.
I have a question regarding the formula $$\int_{0}^{\infty}\int_{0}^{\infty}\sin(xy)xf(x)\,\mathrm dx\,\mathrm dy=\int_{0}^{\infty}f(x)\,\mathrm dx.$$ I derived it this by moving integrals inside each ...
- 1,376
0 votes
0 answers
52 views
Divergence/convergence of improper integrals
The integral $\int_{0}^{\infty} \cos(x) dx$ is considered to be divergence, as well as $ \int_{1}^{\infty} x^{-1} dx$. Now I'm studying curves in space, and there are many integrals from curvature, ...
- 897