Integrate the function w.r.t. x: ( ^ -1 x ) 1+x^ 2 - Vidyadip

Question

Integrate the function $w.r.t. x: \frac{\sin \left(\tan ^{-1} x\right)}{1+x^{2}}$

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Answer

Derivative of $\tan ^{-1} x=\frac{1}{1+x^{2}}$.
Thus, we use the substitution $\tan^{-1} x = t$
so that $\frac{d x}{1+x^{2}}=d\ t$
Therefore, $\int \frac{\sin \left(\tan ^{-1} x\right)}{1+x^{2}} d\ x$
$=\int \sin t\ d\ t = -\cos t + C$
$ = -\cos(\tan^{-1}x) + C$

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