Question Detail

The denominator x2 - x - 2 can be factored as (x-2)(x+1). We use partial fraction decomposition: (4x-5)⁄((x-2)(x+1)) = A⁄(x-2) + B⁄(x+1). Multiplying by the common denominator, we get 4x-5 = A(x+1) + B(x-2). Setting x=2, we get 4(2)-5 = A(2+1) ⇒ 3 = 3A ⇒ A=1. Setting x=-1, we get 4(-1)-5 = B(-1-2) ⇒ -9 = -3B ⇒ B=3. So the integral becomes ∫(1⁄(x-2) + 3⁄(x+1)) dx. Integrating each term gives ln|x-2| + 3ln|x+1| + C.