Question Detail

First, factor the denominator: x2 - x - 2 = (x-2)(x+1).Next, use partial fraction decomposition:(4x - 5) / ((x-2)(x+1)) = A/(x-2) + B/(x+1)Multiply both sides by (x-2)(x+1):4x - 5 = A(x+1) + B(x-2)To find A, set x = 2:4(2) - 5 = A(2+1) + B(0)8 - 5 = 3A3 = 3A => A = 1To find B, set x = -1:4(-1) - 5 = A(0) + B(-1-2)-4 - 5 = -3B-9 = -3B => B = 3So, the integral becomes:∫ [1/(x-2) + 3/(x+1)] dxIntegrate each term:∫ 1/(x-2) dx = ln|x-2| + C1∫ 3/(x+1) dx = 3ln|x+1| + C2Combining them, we get:ln|x-2| + 3ln|x+1| + CThis matches option B (rearranged).