Question Detail

First, factor the denominator: x2 - x - 2 = (x-2)(x+1). Now, 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. Integrating term by term: ∫ (1/(x-2)) dx = ln|x-2| + C, and ∫ (3/(x+1)) dx = 3ln|x+1| + C. Combining them, we get ln|x-2| + 3ln|x+1| + C.