Question

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

Answer 1

`1/2arctan(e^(2x))+C`

Explanation:

`inte^(2x)/(1+e^(4x))dx`

Let `u=e^(2x)` so `du=2e^(2x)dx`.

`=1/2int(2e^(2x))/(1+(e^(2x))^2)dx`

`=1/2int1/(1+u^2)du`

This is the arctangent integral:

`=1/2arctan(u)+C`

`=1/2arctan(e^(2x))+C`

Another way to show this is to use the trigonometric substitution `e^(2x)=tan(theta)`.