Question
Evaluate : $\int_{-a}^a \sqrt{\frac{a-x}{a+x}} d x$
$\int_{-a}^a \sqrt{\frac{a-x}{a+x}} d x$
Let $I=\int_{-a}^a \sqrt{\frac{a-x}{a+x}} d x$
$=\int_{-a}^a \sqrt{\frac{(a-x)(a-x)}{(a+x)(a-x)}} d x$
$=\int_{-a}^a \frac{a-x}{\sqrt{a^2-x^2}} d x$
$=\int_{-a}^a \frac{a}{\sqrt{a^2-x^2}} d x-\int_{-a}^a \frac{x}{\sqrt{a^2-x^2}} d x$
[but $\frac{a}{\sqrt{a^2-x^2}}$ is an is an even function and $\frac{x}{\sqrt{a^2-x^2}}$ is an odd function]
$=2 a \cdot\left[\sin ^{-1}\left(\frac{x}{a}\right)\right]_0^a$
$=2 a \cdot\left[\sin ^{-1} 1-\sin ^{-1} 0\right]$
$=2 a\left[\frac{\pi}{2}-0\right]$
$\int_{-a}^a \sqrt{\frac{a-x}{a+x}} \cdot d x=\pi a$
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