Evaluate : \(\displaystyle\int \frac{2x + 1}{x(x - 1)(x - 4)}\,dx\)

Answer: \(\dfrac{1}{4}\log|x| - \log|x - 1| + \dfrac{3}{4}\log|x - 4| + c\)

Let \(\dfrac{2x + 1}{x(x - 1)(x - 4)} = \dfrac{A}{x} + \dfrac{B}{x - 1} + \dfrac{C}{x - 4}\), so \(2x + 1 = A(x - 1)(x - 4) + Bx(x - 4) + Cx(x - 1)\).

• \(x = 0\): \(1 = 4A \Rightarrow A = \dfrac{1}{4}\)
• \(x = 1\): \(3 = -3B \Rightarrow B = -1\)
• \(x = 4\): \(9 = 12C \Rightarrow C = \dfrac{3}{4}\)

\(\displaystyle\int \frac{2x + 1}{x(x - 1)(x - 4)}\,dx = \frac{1}{4}\int \frac{dx}{x} - \int \frac{dx}{x - 1} + \frac{3}{4}\int \frac{dx}{x - 4}\)

\(= \dfrac{1}{4}\log|x| - \log|x - 1| + \dfrac{3}{4}\log|x - 4| + c\)