Evaluate: _ -a ^ a a - x a + x dx - Vidyadip

Question

$\text{Evaluate:} \int\limits_{-a}^{a} \sqrt\frac{{a - x}}{a + x} {dx}$

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Answer

$\text{I} = \int\limits_{-a}^{a} \sqrt\frac{{a - x}}{a + x} {dx} = \int\limits_{-a}^{a}\frac{a - x}{\sqrt{a^{2} - x^{2}}} dx = \int\limits_{-a}^{a} \frac{x dx}{\sqrt{a^{2} - x^{2}}}$$I_{1}$ is even function and $I_{2}$ is odd function
$\therefore I_{2} = 0$
$i = 2a \int\limits_{-a}^{a} \frac{dx}{\sqrt{a^{2} - x^{2}}} = 2a. \frac{\pi}{2} = \pi\text{a}$
$\therefore I = \pi\text{a}$

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