Linked Questions

While browsing on Integral and Series, I found a strange integral posted by @Sangchul Lee. His post doesn't have a response for more than a month, so I decide to post it here. I hope he doesn't mind ...

I'm a sucker for exotic integrals like the one evaluated in this post. I don't really know why, but I just can't get enough of the amazing closed forms that some are able to come up with. So, what ...

In this question, user Franklin Pezzuti Dyer gives the following surprising integral evaluation: $$\int_0^{\pi/2}\ln \lvert\sin(mx)\rvert \cdot \ln \lvert\sin(nx)\rvert \, dx = \frac{\pi^3}{24} \frac{...

I noticed that a lot of commonly-used mathematical constants that can't be expressed in closed-form can be expressed by integrals, such as $$\pi=\int_{-\infty}^\infty \frac{dx}{x^2+1}$$ and $$\frac{1}{...

I am in the last year of my school and studied integration this year I have done several Integration techniques like Integration By substitution By partial fractions By parts Trigo. substitutions ...

Please let me know if this question does not belong here. I have always wondered how the problem-setters for contests like the IMO come up with the problems. The creation of problems like those set at ...

I know the following result thanks to the technique "Integral Milking": $$\int_0^\infty \frac{\sin(2x)}{1-e^{2\pi x}} dx = \frac{1}{2-2e^2}$$ So I have a proof (I might list it here later, ...

I recently found that $$\int_{-\infty}^{\infty}\frac{\mathrm{d}x}{ax^2+bx+c}=\pi$$ iff $$b^2-4ac=-4.$$ I found it by integrating $$I=\int_{-\infty}^{\infty}\frac{\mathrm{d}x}{ax^2+bx+c}.$$ If the ...

While milking the integral $\int_0^\pi\sin^{-1}\left(\sin x\right)dx$, I believe I may have conceived of a nice proof of $$\sum_{n=1}^\infty\frac{1}{n^2}=\frac{\pi^2}{6}$$ It makes use of the ...

For everything on this post $n$ and $m$ are positive integers. The other day I found the following integral on the popular post "Integral Milking" and decided to give it a go. $$\large\int_{...

Recently, in a pre-calculus textbook I saw lying around my high school, I saw a problem having something to do with the neat identity $$\left(a+\frac1a\right)\left(b+\frac1b\right)\left(ab+\frac1{ab}...

I've been going crazy with this complex integral I have to estimate with the Residues Theorem. I'm obviously missing a sign or something else, but I fear I may be committing a conceptual mistake. $$\...

Where $a,b$ and $c$ are consecutive arithmetic terms. We wish to evaluate this integral, $$\int_{-\infty}^{\infty}\frac{x(x+a)(x+b)(x+c)}{(x^3+2x^2-x-1)^2+(x^2+x-1)^2}\mathrm dx$$ I don't even ...

I found the following integral in chapter $13$ of Irresistible Integrals, and I would like to see which conclusions you can reach from it. My goal in asking this question is to see which methods I can ...

Is there a non-trivial integral yielding $e$? Similar questions have been asked here and here. However, both of these posts asks for integrals yielding elementary functions of $e$, and not necessarily ...