Question

Integral of 1 divided by x square - 2 x

Answer 1

Step-by-step explanation:

We have,

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

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

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

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

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

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

$\tt = \frac{1}{2} \int \frac{1}{ x- 2 } dx- \frac{1}{2} \int \frac{1}{x} dx$

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

$= \tt\frac{1}{2} ln(x- 2) - \frac{1}{2}ln(x) + C$

$= \tt ln \sqrt{x- 2} - ln \sqrt{x} + C$

$= \tt ln \sqrt{ \frac{x- 2}{x}} + C$