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) + 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:∫ 1/(x-2) dx = ln|x-2|∫ 3/(x+1) dx = 3ln|x+1|Thus, the simplified integral is ln|x-2| + 3ln|x+1| + C.This matches option B.