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

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