Question

How do you find the indefinite integral of `int (x^2-6x-20)/(x+5)`?

Answer 1

`int frac (x^2-6x-20) (x+5) dx = (x+5)^2/2 -16(x+5) +35 ln (x+5)`

Explanation:

Substitute `x=t-5`

`int frac (x^2-6x-20) (x+5) dx = int (dt)/t((t-5)^2 -6(t-5) -20) =`

`= int (dt)/t(t^2-10t +25 -6t+30 -20) =`

`= int (dt)/t(t^2-16t +35) = int t dt -16 int dt +35 int (dt)/t =`

`= (t^2)/2 -16t +35 ln(t) = (x+5)^2/2 -16(x+5) +35 ln (x+5)`