Question

How do you find `int (5x^2+3x-2)/(x^3+2x^2) dx` using partial fractions?

Answer 1

`2 ln x + 3 ln (x+2) +1/x+C`

Explanation:

Assume that you can write the fraction as `(Ax+B)/x^2+C/(x+2)`.
That is, `5x^2+3x-2-=(Ax+B)(x+2)+Cx^2`

Then, equating coefficents (or however you prefer)
you get `5=A+C`, `3=2A+B` and `-2=2B`. Hence `B=-1`, `A=2` and `C=3`
So now you have `int(2x-1)/x^2+3/x+2 dx`
=`int2/x-1/x^2+3/(x+2)dx` giving the answer above.