Integrate the rational function: x (x-1)(x-2)(x-3) - Vidyadip

Question

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

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Answer

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) ......(i)
Substituting x = 1, 2 and 3 respectively in equation (1), we get,
$A=\frac{1}{2}, B=-2$ and $C=\frac{3}{2}$
Thus,
$\frac{x}{(x-1)(x-2)(x-3)}=\frac{1}{2(x-1)}-\frac{2}{(x-2)}+\frac{3}{2(x-3)}$
$\Rightarrow$$\int \frac{x}{(x-1)(x-2)(x-3)} d x=\int\left\{\frac{1}{2(x-1)}-\frac{2}{(x-2)}+\frac{3}{2(x-3)}\right\} d x$
= $\frac{1}{2} \log |x-1|-2 \log |x-2|+\frac{3}{2} \log |x-3|+C$

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