# How do you find the antiderivative of (e^(2x))/(1+(e^(4x))dx?

Dec 12, 2016

$\frac{1}{2} \arctan \left({e}^{2 x}\right) + C$

#### Explanation:

$\int {e}^{2 x} / \left(1 + {e}^{4 x}\right) \mathrm{dx}$

Let $u = {e}^{2 x}$ so $\mathrm{du} = 2 {e}^{2 x} \mathrm{dx}$.

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

$= \frac{1}{2} \int \frac{1}{1 + {u}^{2}} \mathrm{du}$

This is the arctangent integral:

$= \frac{1}{2} \arctan \left(u\right) + C$

$= \frac{1}{2} \arctan \left({e}^{2 x}\right) + C$

Another way to show this is to use the trigonometric substitution ${e}^{2 x} = \tan \left(\theta\right)$.