Question

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

Answer 1

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

We can write,

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

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

$\therefore x=A(x-2)+B(x-1)$ ............ $\left(1\right)$

Putting $x=1$ in $\left(1\right)$

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

Putting $a=2$ in $\left(1\right)$

$2=0+B\left(1\right)\Rightarrow B=2$

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

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

$=-\log |x-1|+2\log |x-2|+C$

where $c=constant$ of integration