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How to calculate: $\int \sqrt{(\cos{x})^2-a^2} \, dx$

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    Maybe you should post your probability problem too.2011-06-27

2 Answers 2

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$\int\sqrt{\cos^2 x-a^2}\;dx =\frac{1}{k} \int \sqrt{1-k^2\sin^2x}\;dx$ where $k=\frac{1}{\sqrt{1-a^2}}$ As this seems to come from a physical problem, introduce limits and look into elliptic integrals of the second kind.

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    Does it makes a difference if the integral is $\int_a^b{\int_0^{2\pi}{\sqrt{y^2(\cos^2{\theta}-1)+1}}\mathrm{d}\theta}\mathrm{d}y$? I thought I can start with the inner integral and simplify it to what I wrote. But, I don't want to do the elliptic integral thing!2011-06-29
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In SWP (Scientific WorkPlace), with Local MAPLE kernel, I got the following evaluation

$\begin{eqnarray*} I &:&=\int \sqrt{\cos ^{2}x-a^{2}}dx \\ &=&-\frac{\sqrt{\sin ^{2}x}}{\sin x}a^{2}\text{EllipticF}\left( \left( \cos x\right) \frac{\text{csgn}\left( a^{\ast }\right) }{a},\text{csgn}\left( a\right) a\right) \\ &&-\text{EllipticF}\left( \left( \cos x\right) \frac{\text{csgn}\left( a^{\ast }\right) }{a},\text{csgn}\left( a\right) a\right) \\ &&+\text{EllipticE}\left( \left( \cos x\right) \frac{\text{csgn}\left( a^{\ast }\right) }{a},\text{csgn}\left( a\right) a\right) F \end{eqnarray*}$

where

$F=\sqrt{\frac{-\cos ^{2}x+a^{2}}{a^{2}}}\sqrt{\cos ^{2}x-a^{2}}\text{csgn}\left( a^{\ast }\right) \frac{a}{-\cos ^{2}x+a^{2}}$

As an example:

$\begin{eqnarray*} \int \sqrt{\cos ^{2}x-2^{2}}dx &=&\frac{\sqrt{\sin ^{2}x}}{\sin x}3\text{EllipticF}\left( \frac{1}{2}\cos x,2\right) \\ &&+\text{EllipticE}\left( \frac{1}{2}\cos x,2\right) \frac{\sqrt{-\cos ^{2}x+4}}{\sqrt{\cos ^{2}x-4}} \end{eqnarray*}$

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    @OsamaGamal In fact, [here it is](http://www.mathworks.com/help/matlab/ref/ellipke.html)2013-05-16