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 by the denominator: 4x-5 = A(x+1) + B(x-2).To find A, set x = 2: 4(2)-5 = A(2+1) + B(0) ⇒ 3 = 3A ⇒ A = 1.To find B, set x = -1: 4(-1)-5 = A(0) + B(-1-2) ⇒ -9 = -3B ⇒ B = 3.So the integral becomes: ∫ (1/(x-2) + 3/(x+1)) dx.Integrating each term:∫ 1/(x-2) dx = ln|x-2|∫ 3/(x+1) dx = 3ln|x+1|Therefore, the solution is ln|x-2| + 3ln|x+1| + C. This matches option B.