f(x) = (x-5)/(x^2+5x+4), express this in partial fractions and hence find the integral of f(x) dx between x=0 and x=2, giving the answer as a single simplified logarithm.

f(x) = (x - 5) / (x + 4)(x + 1) let f(x) = A / (x + 4) + B / (x + 1) so (x - 5) / (x + 4)(x + 1) = A / (x + 4) + B / (x + 1) then multiplying by (x+4)(x+1), we get x - 5 = A(x + 1) + B(x + 4) x - 5 = Ax + A + Bx + 4B x - 5 = x(A + B) + A + 4B we can now compare coefficients: 1 x on the left, (A + B)x on the right, so A + B = 1 -5 on the left, (A + 4B) on the right, so A + 4B = -5 solving these simultaneous equations, B = -2 A = 3 so f(x) = 3 / (x + 4) - 2 / (x + 1) then integrating this, we get integral = 3 ln(x + 4) - 2 ln(x + 1) integral at 2 = 3 ln(6) - 2 ln(3) = ln(6^3) - ln(3^2) = ln(216) - ln(9) = ln(216 / 9) = ln(24) integral at 0 = 3 ln(4) - 2 ln(1) = ln(4^3) - 0 = ln(64) so answer = integral at 2 - integral at 0 = ln(24) - ln(64) = ln(24/64) = ln(3/8)