Originally Posted by

**Deveno** no, this is not anything to do with partial fractions.

if we are going to integrate something like (ax+b)/(cx^{2} + dx + e)

it would be a lot easier if the top was the derivative of the bottom. because then we would have log|(something)| when we integrate.

well the derivative of x^{2} - 2x - 3 is 2x - 2. so 2x - 2 is what we "wish" we had in the numerator. well we can make x-4 look "more" like 2x - 2, by multiplying the top and bottom of our fraction by 2, and taking the 2 in the denominator outside the integral:

$\displaystyle \int \frac{x - 4}{x^2 - 2x - 3} dx = \frac{1}{2}\int \frac{2x - 8}{x^2 - 2x - 3} dx$.

now we have 2x - 8 in the top, which we split into the sum 2x - 2 - 6 (so now we have 2 integrals, with the same denominator):

$\displaystyle =\frac{1}{2}\int \frac{2x - 2}{x^2 - 2x - 3} dx - \frac{1}{2}\int \frac{6}{x^2 - 2x - 3} dx $

in the second integral, we can bring the 6 outside the integral, where it cancels with the 1/2 to make 3.