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
Questions 203525
Subjects 9
Chapters 126
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