Question Detail

To simplify the integral ∫ (4x-5) / (x2 - x - 2) dx, use partial fraction decomposition.First, factor the denominator: x2 - x - 2 = (x-2)(x+1).Set up the partial fraction form: (4x-5) / ((x-2)(x+1)) = A/(x-2) + B/(x+1).Multiply by the common denominator:4x-5 = A(x+1) + B(x-2).To find A, set x = 2:4(2)-5 = A(2+1) + B(0) => 8-5 = 3A => 3 = 3A => A = 1.To find B, set x = -1:4(-1)-5 = A(0) + B(-1-2) => -4-5 = -3B => -9 = -3B => B = 3.So the integral becomes:∫ [1/(x-2) + 3/(x+1)] dx= ∫ 1/(x-2) dx + ∫ 3/(x+1) dx= ln|x-2| + 3ln|x+1| + C.This matches option B (order of terms does not matter).