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 = \(\frac{3}{2}\)

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

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

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