Evaluate 1 x(x-1) d x - Vidyadip

Question

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

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Answer

$\text { Let } I =\int \frac{1}{x(x-1)} d x$
$=\int \frac{x-x+1}{x(x-1)} d x$
$=\int \frac{x-(x-1)}{x(x-1)} d x$
$=\int\left(\frac{1}{x-1}-\frac{1}{x}\right) d x$
$=\int \frac{1}{x-1} d x-\int \frac{1}{x} d x$
$=\log |x-1|-\log |x|+ c$
$\therefore I =\log \left|\frac{x-1}{x}\right|+ c $

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