You are correct all the way until , which should be .
The equation is integral sign ("S") (x^3 -4x^2 + 3x + 7)/(x^2 -3x +2) dx
So first I did the long division and I got
the original equation = S (x - 1 + (9-2x)/(x^2-3x+2)) dx
so 9-2x = A + B
____ __ ___
(x-1)(x-2) (x-1) (x-2)
9-2x = A (x-2) + B(x-1)
so if x = 1 then A = -7 and if x = 2 then B = 5
which gives us
S (x-1- 7/(x-2) + 5/(x-1)) dx = (1/2)x^2 -x -7 ln(x-2) + 5 ln (x-1) +C
Did I do this correctly? I don't like ln but I am assuming I bring the costant infront?