Question

Evaluate $\int \frac{1}{(x-1)x^2}dx$

Answer 1

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

Splitting the above expression in to partial fractions, this can be written as

$\Rightarrow \int (\frac{1}{x-1}-\frac{1}{x}-\frac{1}{x^2})dx$

$\Rightarrow \int \frac{1}{x-1}dx-\int \frac{1}{x}dx-\int \frac{1}{x^2}dx$

$\Rightarrow \log (x-1)-\log x+\frac{1}{x}$

$\Rightarrow \log \left(\frac{x-1}{x}\right)+\frac{1}{x}$