Question

Evaluate:

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

Answer 1

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

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

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

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

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

= `int [((e^x - 1)(e^x + 1))/(e^x + 1) + (e^-x)/(1 + e^-x)] dx`

= `int [(e^x - 1) + (e^-x)/(1 + e^-x)] dx`

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

= `e^x - x - log |1 + e^-x| + c     ...[because d/(dx) (1 + e^-x) = -e^-x and int (f'(x))/(f(x))dx = log f(x) + c]`