Question

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

Answer 1

Consider the given integral.

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

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

$I=\frac{1}{2}\int \frac{2}{x}dx-\frac{1}{2}\int \frac{3}{(2x-1)}dx+\frac{1}{2}\int 1dx$

$I=\frac{1}{2}\int \frac{2}{x}dx-\frac{3}{4}\int \frac{1}{(x-\frac{1}{2})}dx+\frac{1}{2}\int 1dx$

$I=\ln x-\frac{3}{4}\ln (x-\frac{1}{2})+\frac{x}{2}+C$