Question

What is the antiderivative of `(x^2-1) / (x^2+1)`?

Answer 1

`x-2 tan^-1 x+C`

Explanation:

`int(x^2-1)/(x^2+1)dx`
`=int(x^2+1-2)/(x^2+1)dx`
`=int1-2/(1+x^2)dx`
`=x-2 tan^-1x+C`.

(If the quoting standard integral `int1/(1+x^2)dx=tan^-1x` is not acceptable, substitute `x=tan u`, `dx=sec^2 u du` and use `sec^2u=1+tan^2u` to reduce the integral to `int 1du`.)