11 questions linked to/from Funny identities
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### A strange integral: $\int_{-\infty}^{+\infty} {dx \over 1 + \left(x + \tan x\right)^2} = \pi.$

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 ...
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### How to prove $\int_0^1\frac{\mathrm{d}x}{x^x}=\sum_{k=1}^\infty \frac1{k^k}$? [duplicate]

I've found this wonderful identity here. On the first spot it seems for me true, but I don't have any idea, how could it be proven. In simpler form: "how the hell...?" :-)
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### How can the following “funny identity” be generalised?

When asked for a "funny identity", Andrey Rekalo answered the following: $$\left(\sum\limits_{k=1}^n k\right)^2=\sum\limits_{k=1}^nk^3 .$$ Not only do I think it's funny, I also think it's very ...
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### On the “funny” identity $\tfrac{1}{\sin(2\pi/7)} + \tfrac{1}{\sin(3\pi/7)} = \tfrac{1}{\sin(\pi/7)}$

This equality in the title is one answer in the MSE post Funny Identities. At first, I thought it had to do with $7$ being a Mersenne prime, but a little experimentation with Mathematica's integer ...
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### How is it possible that $\infty!=\sqrt{2\pi}$?

I read from here that: $$\infty!=\sqrt{2\pi}$$ How is this possible ? $$\infty!=1\times2\times3\times4\times5\times\ldots$$ But \begin{align} 1&=1\\ 1\times2&=2\\ 1\times2\times3&=6\\ &...
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### Surprising identities / equations

What are some surprising equations/identities that you have seen, which you would not have expected? This could be complex numbers, trigonometric identities, combinatorial results, algebraic results, ...
I need to prove the following identity: $\sin^2 2\alpha-\sin^2 \alpha = \sin 3\alpha \sin \alpha$ What I have tried, is to work on each side of the identity. I have started with the left side: \...