Evaluate : \(\displaystyle\int\frac{3x^2 + 4x - 5}{(x^2 - 1)(x + 2)}\cdot dx\)

Answer: \(\dfrac{1}{3}\log|x - 1| + 3\log|x + 1| - \dfrac{1}{3}\log|x + 2| + c\)

\((x^2 - 1)(x + 2) = (x - 1)(x + 1)(x + 2)\). Let

\(\dfrac{3x^2 + 4x - 5}{(x - 1)(x + 1)(x + 2)} = \dfrac{A}{x - 1} + \dfrac{B}{x + 1} + \dfrac{C}{x + 2}\)

\(3x^2 + 4x - 5 = A(x + 1)(x + 2) + B(x - 1)(x + 2) + C(x - 1)(x + 1)\).

• \(x = 1\): \(2 = 6A \Rightarrow A = \dfrac{1}{3}\)
• \(x = -1\): \(-6 = -2B \Rightarrow B = 3\)
• \(x = -2\): \(-1 = 3C \Rightarrow C = -\dfrac{1}{3}\)

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

\(= \dfrac{1}{3}\log|x - 1| + 3\log|x + 1| - \dfrac{1}{3}\log|x + 2| + c\).