# How do I find the integral intdx/(x^2(x-1)^2) ?

Sep 28, 2014

By Partial Fraction Decomposition,

$\frac{1}{{x}^{2} {\left(x - 1\right)}^{2}} = \frac{2}{x} + \frac{1}{x} ^ 2 + \frac{2}{x - 1} + \frac{1}{{\left(x - 1\right)}^{2}}$.

$\int \frac{\mathrm{dx}}{{x}^{2} {\left(x - 1\right)}^{2}}$

by the partial fractions above,

$= \int \frac{2}{x} + \frac{1}{x} ^ 2 + \frac{2}{x - 1} + \frac{1}{{\left(x - 1\right)}^{2}} \mathrm{dx}$

by Log Rule and Power Rule,

$= 2 \ln | x | - \frac{1}{x} + 2 \ln | x - 1 | - \frac{1}{x - 1} + C$