Question

integrate 1 upon x(x-1) dx​

Answer 1

Step-by-step explanation:

We have,

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

$= \int \frac{dx}{ {x}^{2} - x }$

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

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

$= \frac{1}{2 \times \frac{1}{2} } ln \: | \frac{(x - \frac{1}{2}) - \frac{1}{2} }{(x - \frac{1}{2} ) + \frac{1}{2} } | + c$

$= ln \: | \frac{x - 1}{x} | + c$

$= ln \: |x - 1| - ln \: |x| + c$