Question

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

Answer 1

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

Let $\frac{x-2}{x(2x-1)}=\frac{A}{x}+\frac{B}{2x-1}$

$x-2=A(2x-1)+Bx$

Let $x=0$

$-2=A(-1)+0\Rightarrow A=2$

Let $x=\frac{1}{2}$

$\frac{1}{2}-2=\frac{B}{2}$ $=\frac{-3}{2}$

$\Rightarrow B=-3$

$\frac{x-2}{x(2x-1)}=\frac{2}{x}-\frac{3}{2x-1}$

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

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

$I=2{\log }_{e}x-\frac{3}{2}lo{g}_e(2x-1)$