Question

Integrate the rational function: $\frac{x}{(x-1)(x-2)(x-3)}$

Answer 1

Let $\frac{x}{(x-1)(x-2)(x-3)}$ =$\frac{A}{(x-1)}+\frac{B}{(x-2)}+\frac{C}{(x-3)}$

x = A(x-2)(X-3)+B(x-1)(x-3)+C(x-1)(x-2) ...(1)

Substituting x = 1, 2, and 3 respectively in equation (1), we obtain

A = $\frac{1}{2}$, B = -2, and C = $\frac{3}{2}$

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

\Rightarrow \int\frac{x}{(x-1)(x-2)(x-3)}dx=\int \bigg\\{\frac{1}{2(x-1)}-\frac{2}{(x-2)}+\frac{3}{2(x-3)}\bigg\\}dx

=$\frac{1}{2}\log\mid x-1 \mid-2\log\mid x-2\mid+\frac{3}{2}\log\mid x-3\mid+C$