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 (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 = ln|x-2| + C1∫ 3⁄(x+1) dx = 3ln|x+1| + C2Combining them: ln|x-2| + 3ln|x+1| + C.This matches option B.