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) => 3 = 3A => A = 1.To find B, set x = -1: 4(-1)-5 = B(-1-2) => -9 = -3B => B = 3.So the integral becomes ∫ (1/(x-2) + 3/(x+1)) dx.Integrate each term: ∫ 1/(x-2) dx + ∫ 3/(x+1) dx = ln|x-2| + 3ln|x+1| + C.This matches option B.