Question Detail

First, factor the denominator: x2 - x - 2 = (x-2)(x+1).Use partial fraction decomposition: (4x-5) / ((x-2)(x+1)) = A/(x-2) + B/(x+1).Multiply 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) => 3 = 3A => A=1.To find B, set x=-1: 4(-1)-5 = B(-1-2) => -9 = -3B => B=3.So the integral becomes ∫ (1/(x-2) + 3/(x+1)) dx.∫ (1/(x-2)) dx = ln|x-2| + C1.∫ (3/(x+1)) dx = 3ln|x+1| + C2.Combining, the result is ln|x-2| + 3ln|x+1| + C. This matches option B.