Guided Example #1 - Inverse Trigonometric Integrals

${\displaystyle \int \frac{e^{x}dx}{1+e^{2}x}=\int \frac{e^{x}dx}{1+{\left(e\right)}^{x^2}}}$

Letting ${\displaystyle u=e^x,du=e^{x}dx}$, this integral becomes
${\displaystyle \int \frac{du}{1+u^2}={tan}^{-1}u+C={tan}^{-1}{e}^x+C}$.