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.
8,267 questions
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Does the integral $\int_{0}^{1} \frac{\ln(1+x)}{\ln(1-x)}\,dx$ have a known closed form? [duplicate]
I am studying the definite integral
$$
I = \int_{0}^{1} \frac{\ln(1+x)}{\ln(1-x)}\,dx .
$$
The integral does converge:
as $x \to 0$, $\ln(1+x) \sim x$ and $\ln(1-x) \sim -x$, so the ratio tends to $-...
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answers
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How far can an infinite number of unit length planks bridge a right-angled gap?
Problem: You are standing at the origin of an infinite flat earth. One quadrant of the earth is gone, leaving an infinite abyss. You have an infinite supply of unit-length zero-width planks. How far ...
4
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1
answer
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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
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4
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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
$$\int_0^\infty \frac{\ln(1+x^2)}{1+...
- 2,958
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6
answers
367
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Seeking other generalisations to the integral $\int_0^{\infty} \frac{\ln \left(x+\frac{1}{x}\right)}{1+x^2}dx$
The integral
$$
\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
$$...
- 33.2k
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Closed form of $\int_0^{\infty} \frac{\sin (\tan x) \cos ^{2n-1} x}{x} d x?$
Being attracted by the answer in the post
$$\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
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4
answers
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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:
$$\boxed{\...
- 33.2k
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Why does Limit Comparison Test with $g(x) = \log x$ give incomplete convergence range?
I'm trying to determine the values of $n$ for which the following integral converges:
$$\int_0^1 \frac{x^n \log x}{(1+x)^2} \, dx$$
The only point of discontinuity is at $x = 0$.
My attempt:
I used ...
- 65
9
votes
3
answers
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views
Generalisation of $\int_0^{\infty} \frac{\tan ^{-1}\left(\tan ^2 x\right)}{x^2}\,dx .$
Being attracted by the result
$$\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
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4
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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 ...
- 1,124
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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
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vote
1
answer
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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
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votes
1
answer
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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
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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
$$
I=\lim_{t\to\infty}\int_{-t}^t \frac{du}{u^2+2}=\lim_{...
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Proof and applicability of an integral formula $\int_{0}^{\infty}\int_{0}^{\infty}\sin(xy)xf(x)\,\mathrm dx\,\mathrm dy$.
I have a question regarding the formula $$\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 ...
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