Question

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

Answer 1

`1/4arctan(e^(2x)/2)+C`.

Explanation:

Let, `I = inte^(2x)/(4+e^(4x))dx`.

We take subst. `e^(2x)=t rArr e^(2x)*2dx=dt`.

Therefore, `I=1/2int(2*e^(2x)*dx)/{4+(e^(2x))^2}`,

`=1/2intdt/(4+t^2) = 1/2*1/2*arctan(t/2)`.

Hence, `I=1/4arctan(e^(2x)/2)+C`.