Mathematics

Evaluate the following integrals
$$\displaystyle\int {\dfrac{{{x^2}}}{{{{\left( {{a^2} - {x^2}} \right)}^{3/2}}}}dx} $$


SOLUTION
put $$x=a sin \theta$$
$$sin \theta =\dfrac{x}{a}$$
$$dx=a\cos \theta d\theta $$
$$\cos \theta \sqrt{1-\dfrac{x^2}{a^2}}=\dfrac{\sqrt{a^2-x^2}}{a}$$
$$I=\displaystyle \int \dfrac{a^2\sin^2\theta \times a \cos \theta d \theta }{(a^2(1-\sin^2\theta ))^{3/2}}$$
$$\int \dfrac{a^3 \sin^2\theta \cos\theta d\theta}{a^3 \cos^3\theta}$$
$$I\int \tan^2\theta d \theta$$
$$I=\int\tan^1\theta d\theta $$
$$I=\displaystyle \int \dfrac{1-\cos^2\theta }{\cos^2\theta }d \theta =\int \sec^2 \theta d \theta -\int d \theta$$
$$=tan \theta -\theta +C$$
$$I=\dfrac{x}{\sqrt{a^2-x^2}}-\sin^{-1}\left(\dfrac{x}{a}\right)+C$$
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Subjective Medium Published on 17th 09, 2020
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