Mathematics

Integrate $$\displaystyle \int \sqrt{\dfrac{a - x}{a + x}} dx$$


SOLUTION
$$\int \sqrt{\dfrac{a-x}{a+x}}dx$$

substitute $$ x=a\cos \theta \rightarrow dx=-a\sin \theta d\theta $$

$$-\int \sqrt{\dfrac{a-a\cos \theta}{a+a\cos \theta}}-a\sin \theta d\theta $$

$$-\int \sqrt{\dfrac{a(2\sin ^2\frac{\theta }{2})}{a(2\cos ^2\frac{\theta }{2})}}-a2\sin \dfrac{\theta }{2}\cos \dfrac{\theta }{2} d\theta $$

$$-2a\int \sin ^2\dfrac{\theta }{2}d\theta $$

$$-2a\int \dfrac{1}{2}(1-\cos \theta )d\theta $$

$$-a[\theta -\sin \theta ]+C$$

$$-a\left [ \cos^{-1}\left ( \dfrac{x}{a} \right )-\sqrt{1-\left ( \dfrac{x}{a} \right )^2} \right ]+C$$
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Subjective Medium Published on 17th 09, 2020
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