Question Detail

First, factor the denominator: x²-x-2 = (x-2)(x+1).Next, decompose the integrand using partial fractions: (4x-5)/((x-2)(x+1)) = A/(x-2) + B/(x+1).Multiply by (x-2)(x+1) to clear denominators: 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, the result is ln|x-2| + 3ln|x+1| + C. This matches option B.