Question Detail

First, factor the denominator: x² - 6x + 8 = (x-4)(x-2). Now, use partial fraction decomposition. (x-1)/((x-4)(x-2)) = A/(x-4) + B/(x-2). Multiply both sides by (x-4)(x-2): x-1 = A(x-2) + B(x-4). To find A, set x = 4: 4-1 = A(4-2) => 3 = 2A => A = 3/2. To find B, set x = 2: 2-1 = B(2-4) => 1 = -2B => B = -1/2. Now, integrate the decomposed fractions: ∫ (3/2)/(x-4) dx + ∫ (-1/2)/(x-2) dx = (3/2) ∫ 1/(x-4) dx - (1/2) ∫ 1/(x-2) dx. This evaluates to (3/2)ln|x-4| - (1/2)ln|x-2| + C.