Question

Integrate:$\int \frac{x}{1-2x}dx$

Answer 1

$I=\int \left(\frac{x}{1-2x}\right)dx=\left(\frac{1}{2}\right)\int \left(\frac{2x}{1-2x}\right)dx=-\left(\frac{1}{2}\right)\int \left(\frac{1-2x-1}{1-2x}\right)dx=-\left(\frac{1}{2}\right)[\int 1dx-\int (\frac{1}{1-2x}\left)dx\right]=-\left(\frac{1}{2}\right)[x-(\frac{log|1-2x|}{-2}\left)\right]+C$