Evaluate the following integrals:

Question:

Evaluate the following integrals:

$\int \frac{\sqrt{1+x^{2}}}{x^{4}} d x$

Solution:

let $x=\tan \theta$, so $d x=\sec ^{2} \theta d \theta$ and $\theta=\tan ^{-} x$

Putting above values,

$=\int \frac{\sqrt{1+x^{2}}}{x^{4}} d x=\int \frac{\sqrt{1+\tan ^{2} \theta}}{\tan ^{4} \theta} \sec ^{2} \theta d \theta=\int \sec ^{2} \theta / \tan ^{2} \theta d \theta$

$=\int \operatorname{cosec}^{2} \theta d \theta=-\cot \theta+c$

Put $\theta=\tan ^{-} \mathrm{x}$

$=-\cot \theta+c=-\cot \tan ^{-} x+c$