Indefinite Integration — Maths STD 12 Science — Question

$=\int a^{x+\tan ^{-1} x} \cdot\left(\frac{x^2+2}{x^2+1}\right) d x$

Put $x+\tan ^{-1} x=t$

$\begin{array}{c}\therefore\left(1+\frac{1}{1+x^2}\right) d x=d t \\ \therefore\left(\frac{1+x^2+1}{1+x^2}\right) d x=d t \\ \therefore\left(\frac{x^2+2}{x^2+1}\right) d x=d t \\ \therefore I=\int a^t d t=\frac{a^t}{\log a}+c \\ =\frac{a^{x+\tan ^{-1} x}}{\log a}+c .\end{array}$