Question Detail

First, factor the denominator: x²-6x+8 = (x-2)(x-4).Now, use partial fraction decomposition:(-x-1) / ((x-2)(x-4)) = A/(x-2) + B/(x-4)Multiply by (x-2)(x-4) to clear denominators:-x-1 = A(x-4) + B(x-2)To find A, let x = 2:-2-1 = A(2-4) => -3 = -2A => A = 3/2To find B, let x = 4:-4-1 = B(4-2) => -5 = 2B => B = -5/2So the integral becomes:∫[(3/2)/(x-2) + (-5/2)/(x-4)] dx= (3/2)∫1/(x-2) dx - (5/2)∫1/(x-4) dx= (3/2)ln|x-2| - (5/2)ln|x-4| + CThis matches option A.