Question:
Evaluate the following integrals:
$\int \frac{1}{\sqrt{1-x^{2}}\left(2+3 \sin ^{-1} x\right)} d x$
Solution:
Assume $2+3 \sin ^{-1} x=t$
$d\left(2+3 \sin ^{-1} x\right)=d t$
$\Rightarrow \frac{3}{\sqrt{1-x^{2}}} d x=d t$
$\Rightarrow \frac{d x}{\sqrt{1-x^{2}}}=\frac{d t}{3}$
But $t=2+3 \sin ^{-1} x$
$=\frac{1}{\mathrm{~b}^{2}} \ln \left|2+3 \sin ^{-1} \mathrm{x}\right|+\mathrm{c}$