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

Question

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

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Answer

$
\begin{aligned}
& \int \frac{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 \\
= & \log \left|\frac{x-1}{x}\right|+c .
\end{aligned}
$

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