Evaluate the following integrals:

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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