Mathematics

# Evaluate :$\displaystyle \int{\dfrac{\sqrt{a^{2}-x^{2}}}{x^{2}}}dx$.

##### SOLUTION
We have,
$\int \dfrac{\sqrt{{a^2}-{x^2}}{x^2}}dx$

Let put $x=a\sin \theta ---(1)$

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

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

$=\int\dfrac {{{a}\sqrt{1\sin^2 \theta}}}{{a^2 \sin^2\theta}}\ a\cos\theta d\theta$

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

$=\int\cot^2 \theta d\theta$

$=\int (\csc^2 \theta-1)d\theta$

$=\int \csc^2 \theta- \int 1 d\theta$

$=-\cot \theta -\theta+ c$

by equation (1)
$x=a \sin \theta$

$\dfrac{x}{a}\ \sin \theta$

$\theta=\sin^{-1}\dfrac{x}{a}$

now,
$-\cot \theta - \theta +c$

$-\cot \sin^{-1}\dfrac{x}{a}-\sin^{-1}\dfrac{x}{a}+c$

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Subjective Medium Published on 17th 09, 2020
Questions 203525
Subjects 9
Chapters 126
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