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

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

Hence this is the answer
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Subjective Medium Published on 17th 09, 2020
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