Question

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

Answer 1

put $x=asin\theta$

$sin\theta =\frac{x}{a}$

$dx=a\cos \theta d\theta$

$\cos \theta \sqrt{1-\frac{x^2}{a^2}}=\frac{\sqrt{a^2-x^2}}{a}$

$I=\int \frac{a^2{\sin }^2\theta \times a\cos \theta d\theta }{\left(a^2\right(1-{\sin }^2\theta ){)}^{3/2}}$

$\int \frac{a^3{\sin }^2\theta \cos \theta d\theta }{a^3{\cos }^3\theta }$

$I\int {\tan }^2\theta d\theta$

$I=\int {\tan }^1\theta d\theta$

$I=\int \frac{1-{\cos }^2\theta }{{\cos }^2\theta }d\theta =\int {\sec }^2\theta d\theta -\int d\theta$

$=tan\theta -\theta +C$

$I=\frac{x}{\sqrt{a^2-x^2}}-{\sin }^{-1}\left(\frac{x}{a}\right)+C$