Almost! the coefficient of ln(x - 3) is just 4, not 4/3

(x+1)/(x^2 - 5x + 6) = (x+1)/(x - 3)(x - 2)

so we have (x+1)/(x - 3)(x - 2) = A/(x - 3) + B/(x - 2)

multiply through by (x - 3)(x - 2) we obtain

x + 1 = A(x - 2) + B(x - 3)

=> x + 1 = Ax - 2A + Bx - 3B

=> x + 1 = (A + B)x + (-2A - 3B)

=> A + B = 1..........................(1)

and -2A - 3B = 1 ....................(2)

2A + 2B = 2 ................(3) = (1)*2

-2A - 3B = 1 ...............(2)

=> -B = 3

=> B = -3

=> A - 3 = 1

=> A = 4

so (x+1)/(x^2 - 5x + 6) = 4/(x - 3) - 3/(x -2)

so int{(x+1)/(x^2 - 5x + 6)}dx = int{4/(x - 3) - 3/(x -2)}dx

= 4ln(x - 3) - 3ln(x - 2) + C ...........so you were almost correct.

Remember, whenever you integrate and get an answer, you can check if you're right by differentiating the answer you got.

d/dx(4ln(x - 3) - 3ln(x - 2) + C) = 4*(1/x - 3) - 3*(1/(x - 2))

= 4/(x - 3) - 3/(x - 2) + 0

= [4(x - 2) - 3(x - 3)]/[(x - 3)(x - 2)]

= (4x - 8 - 3x + 9)/(x - 3)(x - 2)

= (x + 1)/(x - 3)(x - 2) ................as desired