# 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 \begin{equation} \sum_{k = 1}^{N-1} \sin(\frac{k}{N}\pi)^{-2} = \frac{N^2 - 1}{3}, \end{equation} 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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$\int \frac x {\sqrt {1-x^2}}dx$ I was attempting to solve this integral, and it would appear the solution to it is $-\sqrt{1 -x^2}+C$. When I attempted to solve it, however, I attempted to let $x = ... 2 votes 1 answer 57 views ### Where does the absolute value come from in this integral evaluation? I have an integral which can be evaluated by trig substitution: $$\int\frac{1}{x\sqrt{x^2-4}} dx.$$ I let$x=2\sec(u)$so that we have $$\int\frac{2\sec(u)\tan(u)}{2\sec(u)\sqrt{4\sec^2(u)-4}} du=\int\... 0 votes 1 answer 43 views ### Axler 6.9: Show that the following list is orthonormal. Let n \in \mathbb{Z_{+}} and show that the list below is an orthonormal list of vectors in C[-\pi,\pi] in the vector space of real valued functions on [-\pi,\pi] with the inner product given ... 3 votes 3 answers 73 views ### How to find the right answer for Integral of \sin(2x)\cos(2x) This is what I did when I solved the$$\int(\sin(2x)\cos(2x))dx$$First I used integration and made$$u=2x$$Then I found the derivative of 2x and determined its value when it is equal to dx,$$\... 1 vote 1 answer 57 views ### Prove$ \int_0^1 (1+a+a^2\pi^2x^{2a})\sin(\pi x^a)dx = a\pi$using previous results If$a>0$prove, $$\int_0^1 \sin(\pi x^a)dx+a\pi\int_0^1 \ x^a\cos(\pi x^a)dx = 0 \tag1$$ $$\int_0^1 \cos(\pi x^a)dx-a\pi\int_0^1 \ x^a\sin(\pi x^a)dx = -1 \tag2$$ $$\int_0^1 (1+a+a^2\pi^2x^{2a})... -4 votes 1 answer 102 views ### \int_{0}^{\pi} \frac{x \tan x}{\sec x \tan x} d x I have solved the Q. Answer in my textbook is -2\pi + x but I got -1. So , I want to confirm my answer and understand where I have done mistake.$$\int_{0}^{\pi} \frac{x \tan x}{\sec x \tan x} d ... 0 votes 0 answers 63 views ### Show that a definite integral is non negative. Consider the definite integral $$I=AB+CD$$ where $$A=\int_{1}^{100}\sinh\left(\frac{\ln x}{8}\right)\cos \left(\frac{\ln x}{2}\right)dx$$$\$ B=\int_{1}^{100}\cosh\left(\frac{\ln x}{8}\right)\cos \... 