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

Question

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

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Answer

Let $\frac{3 x-1}{(x-1)(x-2)(x-3)}=\frac{A}{(x-1)}+\frac{B}{(x-2)}+\frac{C}{(x-3)}$
$\Rightarrow$ 3x -1 = 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 (i), we get,
A = 1, B = -5 and C = 4
Thus,
$\frac{3 x-1}{(x-1)(x-2)(x-3)}=\frac{1}{(x-1)}-\frac{5}{(x-2)}+\frac{4}{(x-3)}$
$\Rightarrow~\int \frac{3 x-1}{(x-1)(x-2)(x-3)} d x=\int\left\{\frac{1}{(x-1)}-\frac{5}{(x-2)}+\frac{4}{(x-3)}\right\} d x$
= log|x - 1| - 5 log|x - 2| + 4 log|x - 3| + C

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