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

##### 1 Answer
Feb 22, 2017

$\int \setminus {e}^{2 x} / \left(1 + {e}^{2 x}\right) \setminus \mathrm{dx} = \frac{1}{2} \ln | 1 + {e}^{2 x} | + c$

#### Explanation:

We want to find $\int \setminus {e}^{2 x} / \left(1 + {e}^{2 x}\right) \setminus \mathrm{dx}$

A trivial substitution can be used to simplify the denominator; Let:

$u = 1 + {e}^{2 x}$

Then $\frac{\mathrm{du}}{\mathrm{dx}} = 2 {e}^{2 x} \implies \frac{1}{2} \frac{\mathrm{du}}{\mathrm{dx}} = {e}^{2 x}$

If we substitute this into the integral we get;

$\int \setminus {e}^{2 x} / \left(1 + {e}^{2 x}\right) \setminus \mathrm{dx} = \int \setminus \frac{\frac{1}{2}}{u} \setminus \mathrm{du}$
$\text{ } = \frac{1}{2} \ln | u | + c$

And if we undo the substitution we get:

$\int \setminus {e}^{2 x} / \left(1 + {e}^{2 x}\right) \setminus \mathrm{dx} = \frac{1}{2} \ln | 1 + {e}^{2 x} | + c$