x d x (x-1)(x-2) equals - Vidyadip

Question

$\int \frac{x d x}{(x-1)(x-2)}$ equals

✓

Answer

Let $\frac{x}{(x-1)(x-2)}=\frac{A}{x-1}+\frac{B}{x-2}$
x = A(x - 2) + B(x - 1) …(i)
Substituting x = 1 and 2 in (i), we get,
A = -1 and B = 2
Therefore, $\frac{x}{(x-1)(x-2)}=-\frac{1}{(x-1)}+\frac{2}{(x-2)}$
$\int \frac{x}{(x-1)(x-2)} d x=\int\left\{-\frac{1}{(x-1)}+\frac{2}{(x-2)}\right\} d x$
= $-\log |x-1|+2 \log |x-2|+c$
= $\log \left|\frac{(x-2)^{2}}{x-1}\right|+C$

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