evaluate integral by substitution (e^x-e^(-x))/(e^x+e^(-x))dx

Answer 1

Note that you have Integral(tanh(x) dx)
= Integral (sinh(x)/cosh(x) dx)
let u = cosh(x)
du = sinh(x) dx

and you have Integral(du/u) = ln(u) = ln(cosh(x)) = ln(e^x + e^-x) + C

Equivalently, let u = e^x + e^-x
Then du = e^x - e^-x du

and you have du/u