Question

How do you evaluate the integral `int e^x/(1+e^(2x))dx` from `-oo` to `oo`?

Answer 1

`pi`

Explanation:

Making `y = e^x` we have `dy = e^x dx = y dx`

`int e^x/(1+e^(2x))dx equiv int dy/(1+y^2) = arctan(y)`

but

`lim_{x->oo}e^x=lim_{y->oo}=oo`

so

`lim_{a->oo} int_{-a}^a dy/(1+y^2) = pi/2-(-pi/2) = pi`