Question:
Evaluate the following integrals:
$\int \frac{\sin \left(\tan ^{-1} x\right)}{1+x^{2}} d x$
Solution:
Assume $\tan ^{-1} x=t$
$d\left(\tan ^{-1} x\right)=d t$
$\Rightarrow \frac{1}{x^{2}+1}=d t$
Substituting $t$ and $d t$
$\Rightarrow \int \sin t d t$
$=-\cos t+c$
But $t=\tan ^{-1} x$
$\Rightarrow-\cos \left(\tan ^{-1} x\right)+c$
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