Question

Integrate the rational function: \(\frac {5x}{(x+1)(x^2-4)}\)

Answer 1

\(\frac {5x}{(x+1)(x^2-4)}\)= \(\frac {5x}{(x+1)(x+2)(x-2)}\)

Let \(\frac {5x}{(x+1)(x+2)(x-2)}\) = \(\frac {A}{(x+1)} +\frac {B}{(x+2)} + \frac {C}{(x-2)}\)

\(5x = A(x+2)(x-2) + B(x+1)(x-2) + C(x+1)(x+2)          ….....(1)\)

Substituting x = −1, −2, and 2 respectively in equation (1), we obtain

A = \(\frac 53\), B= \(-\frac {5}{2}\), and C = \(\frac 56\)

∴ \(\frac {5x}{(x+1)(x+2)(x-2)}\) = \(\frac {5}{3(x+1)} -\frac {5}{2(x+2)} + \frac {5}{6(x-2)}\)

⇒ \(∫\)\(\frac {5x}{(x+1)(x+2)(x-2)} \ dx\) = \(\frac 53 ∫\frac {1}{(x+1)}dx-\frac 52 ∫\frac {1}{(x+2)}dx+\frac 56 ∫\frac {1}{(x-2)}dx\)

= \(\frac 53\ log|x+1|-\frac 52\ log|x+2|+\frac 56\ log|x-2|+C\)