Question Detail

First, factor the denominator: x²-x-2 = (x-2)(x+1).Next, use partial fraction decomposition to rewrite the integrand:(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) &implies; 3 = 3A &implies; A=1.To find B, set x=-1: 4(-1)-5 = B(-1-2) &implies; -9 = -3B &implies; B=3.So the integral becomes:∫ (1/(x-2) + 3/(x+1)) dxIntegrate each term:∫ (1/(x-2)) dx + ∫ (3/(x+1)) dx = ln|x-2| + 3ln|x+1| + C.This matches option B after reordering the terms.