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 (x-2)(x+1) to get: 4x-5 = A(x+1) + B(x-2).Set x = 2: 4(2)-5 = A(2+1) => 3 = 3A => A = 1.Set x = -1: 4(-1)-5 = B(-1-2) => -9 = -3B => B = 3.So the integral becomes ∫ [1/(x-2) + 3/(x+1)] dx.Integrate each term: ∫ 1/(x-2) dx = ln|x-2| and ∫ 3/(x+1) dx = 3ln|x+1|.Combining these gives: ln|x-2| + 3ln|x+1| + C.