Integration by parts - help!!!?

Question

The question is: integration the following equation using integration by parts only:

(2x+1)/(x^2+x-2)

Thanks!

Answer 1

ʃ(2x + 1)dx / (x² + x - 2)

Let x² + x - 2 = t

Diferenciate each side with respective x

Recall : Derivative of xⁿ = n (x^n-1)

(2x + 1)dx = dt

Substitute x² + x - 2 = t and (2x + 1) dx = dt

ʃ2x + 1)dx / (x² + x - 2) = ʃdt / t

                                        = log|t| + C

Substitute t = x² + x  - 2

                                        = log|x² + x - 2| + C.