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

Aug 3, 2016

$\frac{1}{4} \arctan \left({e}^{2 x} / 2\right) + C$.

#### Explanation:

Let, $I = \int {e}^{2 x} / \left(4 + {e}^{4 x}\right) \mathrm{dx}$.

We take subst. ${e}^{2 x} = t \Rightarrow {e}^{2 x} \cdot 2 \mathrm{dx} = \mathrm{dt}$.

Therefore, $I = \frac{1}{2} \int \frac{2 \cdot {e}^{2 x} \cdot \mathrm{dx}}{4 + {\left({e}^{2 x}\right)}^{2}}$,

$= \frac{1}{2} \int \frac{\mathrm{dt}}{4 + {t}^{2}} = \frac{1}{2} \cdot \frac{1}{2} \cdot \arctan \left(\frac{t}{2}\right)$.

Hence, $I = \frac{1}{4} \arctan \left({e}^{2 x} / 2\right) + C$.