Integrate the function $\frac{e^{\tan ^{-1} x}}{1+x^{2}}$.

(N/A) Let $I = \int \frac{e^{\tan ^{-1} x}}{1+x^{2}} dx$.
Substitute $\tan ^{-1} x = t$.
Differentiating both sides with respect to $x$,we get $\frac{1}{1+x^{2}} dx = dt$.
Substituting these into the integral,we get $\int e^{t} dt$.
The integral of $e^{t}$ is $e^{t} + C$.
Substituting back $t = \tan ^{-1} x$,we get the final result as $e^{\tan ^{-1} x} + C$,where $C$ is an arbitrary constant.