Question Detail

To simplify the integral ∫ (4x-5) / (x²-x-2) dx, 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) to clear the denominators: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 = 3A3 = 3A => A = 1To find B, set x=-1:4(-1)-5 = A(-1+1) + B(-1-2)-4-5 = -3B-9 = -3B => B = 3So the integral becomes:∫ (1/(x-2) + 3/(x+1)) dxIntegrate each term:∫ (1/(x-2)) dx = ln|x-2| + C₁∫ (3/(x+1)) dx = 3ln|x+1| + C₂Combining these, the result is: ln|x-2| + 3ln|x+1| + C.This matches option B.