Question

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

Answer 1

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

`int1/(e^x+e^(-x))dx`

Start by multiplying numerator and denominator by `e^x`

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

Substitute `t = e^x`

`dt = e^xdx`

`int1/(t^2+1)`

We can see the derivative of `arctan(t)`

`[arctan(t)]+C`

Substitute back

`arctan(e^x)+C`