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 common denominator (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)3 = 3A => A = 1To find B, set x = -1:4(-1)-5 = A(0) + B(-1-2)-9 = -3B => B = 3So the integral becomes:∫ (1/(x-2) + 3/(x+1)) dxIntegrate each term (using ∫ 1/u du = ln|u|):= ln|x-2| + 3ln|x+1| + C.This matches option B (the order of terms can vary).