# Questions tagged [trigonometric-integrals]

Relating to integrations consisting of only(mainly) trigonometric functions and/or requiring substitutions by/of trigonometric functions.

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### On the Fourier series expansion of $\sin(\pi x)$ periodic on $(-\frac{1}{2}, -\frac{1}{2})$

For an Undergrad. sophomore Math Methods class I am taking this session, we have recently covered Fourier series expansions. I made good progress in one of the exercises, but I was stuck for days on ...
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### Integrals, and even and odd functions

I am looking at a proof and got stuck on a part with an integral. I tried to simplify the problem as much as possible, I hope I did not omit any potential helpful information. I have an even ...
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### What type of trigonometric integral is $\int_{-\pi}^{\pi} \sin^{2}(x)\sin(n\pi x)\,dx$?

None of the trigonometric integrals look like the integral I have written. I need use some type identity trigonometric ? $$\int_{-\pi}^{\pi} \sin^{2}(x)\sin(n\pi x)\,dx$$
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### How to prove that $\sum_{ k= 1}^{N-1} \sin(\frac{k}{N} \pi)^{-2} = (N^2 - 1)/3$? [duplicate]

I find the equality $$\sum_{k = 1}^{N-1} \sin(\frac{k}{N}\pi)^{-2} = \frac{N^2 - 1}{3},$$ during study. Just wonder how can I prove it? Short matlab code to do the ...
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### Integrate $\int_0^{\infty} \frac{\sin^2 x}{\cosh x\>+\>\cos x}\frac{dx}x$

It is known that (see for example) \begin{align} &\int_0^{\infty} \frac{\sin x}{\cosh x+\cos x}\frac{dx}x =\frac\pi4\\ &\int_0^{\infty} \frac{\sin^3 x}{\cosh x+\cos x}\frac{dx}x =\frac\pi8 \...
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