Question Detail

First, factor the denominator: x² - x - 2 = (x - 2)(x + 1).Use partial fraction decomposition: (x-1)/((x-2)(x+1)) = A/(x-2) + B/(x+1).Multiplying by (x-2)(x+1): x - 1 = A(x + 1) + B(x - 2).Set x = 2: 1 = A(3) => A = 1/3.Set x = -1: -2 = B(-3) => B = 2/3.So the integral is ∫ [ (1/3)/(x-2) + (2/3)/(x+1) ] dx = (1/3) ln |x-2| + (2/3) ln |x+1| + C.This expression can be written as ln |(x-2)^(1/3) (x+1)^(2/3)| + C. This does not directly match option D, which is (2/3) ln |(x-2)(x+1)| = (2/3) ln |x-2| + (2/3) ln |x+1|. There is a discrepancy in the coefficients for ln|x-2| between the derived answer (1/3) and the highlighted option (2/3).