# 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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### Integral $\int_0^{\frac{\pi}{2}}\frac{\log\left(\sin x\right)}{\cos^2x+y^2\sin^2x}{d}x=-\frac{\pi}{2}\frac{\log\left(1+y\right)}{y}$

Prove that $$\int_0^{\frac{\pi}{2}}\frac{\log\left(\sin x\right)}{\cos^2\left(x\right)+y^2\sin^2\left(x\right)}{\rm d}x=-\frac{\pi}{2}\frac{\log\left(1+y\right)}{y}$$ where $y\ge0$. I came across this ...
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### examples of trigonometric functions $f(x)$ such that $\int_0^M e^{if(x)} dx$ becomes a desired value

Let $f(x)$ be a non-constant differentiable real-valued function defined on $x \in [0,M]$. What are some examples of trigonometry functions $f(x)$ such that it satisfies the following equations: \...
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### How to differentiate a trig value? [closed]

This is a a function concerning the differentiation of a trig value $\sin^{2}{x}$ Does anyone knows what is its derivative? $g(x)=\sin^{2}((\frac{i}{(1-3x)})^{\frac{1}{2}})$
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### Prove $\int_0^\pi\arcsin(\frac14\sqrt{8-2\sqrt{10-2\sqrt{17-8\cos x}}})dx=\frac{\pi^2}{5}$.

There is numerical evidence that $$\int_0^\pi\arcsin\left(\frac14\sqrt{8-2\sqrt{10-2\sqrt{17-8\cos x}}}\right)dx=\frac{\pi^2}{5}.$$ How can this be proved? Context In another question, three random ...
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### Prove $\int_0^1\frac{1}{\sqrt{1-x^2}}\arccos\left(\frac{3x^3-3x+4x^2\sqrt{2-x^2}}{5x^2-1}\right)\mathrm dx=\frac{3\pi^2}{8}-2\pi\arctan\frac12$.

There is numerical evidence that $$I=\int_0^1\frac{1}{\sqrt{1-x^2}}\arccos\left(\frac{3x^3-3x+4x^2\sqrt{2-x^2}}{5x^2-1}\right)\mathrm dx=\frac{3\pi^2}{8}-2\pi\arctan\frac12.$$ How can this be proved? ...
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### How to solve this integral trigonometric integral or prove that is does not have a closed form solution? [closed]

How would I solve this. $$I= \int_0^{\frac{\pi}{4}} \sqrt{\sec^2(x)+\sin^2(x)}\ dx$$ The approximate value of the integral: $$I = 0.937723050585$$ I am trying to solve this integral to find the arc ...
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