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### Find the average of $\sin^{100} (x)$ in 5 minutes?

I read this quote attributed to VI Arnold. "Who can't calculate the average value of the one hundredth power of the sine function within five minutes, doesn't understand mathematics - even if he ...
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### Purpose Of Adding A Constant After Integrating A Function

I would like to know the whole purpose of adding a constant termed constant of integration everytime we integrate an indefinite integral $\int f(x)dx$. I am aware that this constant "goes away" when ...
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### Tricky integration?

How would you compute for the definite integral of $$\int_0^{\infty}\frac{dx}{(1+x^2)^4}$$ I know that integral of $\displaystyle \frac1{(1+x^2)}$ equals $\tan^{-1}x$. I tried using integration by ...
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### Closed form for integral $\int_{0}^{\pi} \left[1 - r \cos\left(\phi\right)\right]^{-n} \phi \,{\rm d}\phi$

Is there a closed form for $$I_n =\int_{0}^{\pi} \frac{\phi}{(1 - r \cos\phi)^n} \,{\rm d}\phi$$ for $\left\vert\,r\,\right\vert < 1$ real and $n > 0$ integer ? The solution to this integral ...
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### Differentiability and decay of magnitude of fourier series coefficients

I want to know the answer/references for the question on decay of Fourier series coefficients and the differentiability of a function. Does the magitude of fourier series coefficients {$a_k$} of a ...
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### numerical evaluation of an integral

I have this integral: $$\int_{-1}^{1} \frac{e^x}{\sqrt{1-x^2}}\,dx$$ How can I get rid of the infinities at the ends of the interval so that I can evaluate this integral numerically? I tried to make ...
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### $\int_0^{\pi/4}\!\frac{\mathrm dx}{2+\sin x}$ , $\int_0^{2\pi}\!\frac{\mathrm dx}{2+\sin x}$

Please help me integrate $$\int_0^{\pi/4}\!\frac{\mathrm dx}{2+\sin x}$$ and $$\int_0^{2\pi}\!\frac{\mathrm dx}{2+\sin x}$$ I've tried the standard $u = \tan \frac{x}{2}$ substitution but it looks ...
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### Computing $\int_0^{\pi\over2} \frac{dx}{1+\sin^2(x)}$?

How would you compute$$\int_0^{\pi\over2} \frac{dx}{1+\sin^2(x)}\, \, ?$$

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