# How do you find the integral of (1/(e^x+e^-x))dx?

May 26, 2015

it's not by part, just a little trick and then substitution

$\int \frac{1}{{e}^{x} + {e}^{- x}} \mathrm{dx}$

Start by multiplying numerator and denominator by ${e}^{x}$

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

Substitute $t = {e}^{x}$

$\mathrm{dt} = {e}^{x} \mathrm{dx}$

$\int \frac{1}{{t}^{2} + 1}$

We can see the derivative of $\arctan \left(t\right)$

$\left[\arctan \left(t\right)\right] + C$

Substitute back

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