Question

\[\int\frac{2x - 1}{\left( x - 1 \right)^2} dx\]

Answer 1

\[\int\left[ \frac{2x - 1}{\left( x - 1 \right)^2} \right]dx\]
\[ = \int\left[ \frac{2x - 2 + 2 - 1}{\left( x - 1 \right)^2} \right]dx\]
\[ = \int\left( \frac{2 \left( x - 1 \right)}{\left( x - 1 \right)^2} + \frac{1}{\left( x - 1 \right)^2} \right)dx\]
\[ = 2\int\frac{dx}{x - 1} + \int \left( x - 1 \right)^{- 2} dx\]
\[ = \text{2 ln }\left| x - 1 \right| + \frac{\left( x - 1 \right)^{- 2 + 1}}{- 2 + 1} + C\]
\[ = \text{2 ln }\left| x - 1 \right| - \frac{1}{x - 1} + C\]