Question Detail

To simplify this integral, we use partial fraction decomposition.First, factor the denominator: x2 - x - 2 = (x - 2)(x + 1).Now, set up the partial fraction decomposition:(4x - 5) / ((x - 2)(x + 1)) = A / (x - 2) + B / (x + 1)Multiply both sides 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) + B(2 - 2)8 - 5 = 3A + 03 = 3A => A = 1To find B, set x = -1:4(-1) - 5 = A(-1 + 1) + B(-1 - 2)-4 - 5 = 0 - 3B-9 = -3B => B = 3So the integral becomes:∫ [1 / (x - 2) + 3 / (x + 1)] dxIntegrate each term:∫ (1 / (x - 2)) dx = ln|x - 2|∫ (3 / (x + 1)) dx = 3 ln|x + 1|Combining these results, the simplified integral is:ln|x - 2| + 3 ln|x + 1| + CThis matches option B.