Question

Evaluate :
$\int \frac{x}{{\left(x-1\right)}^2\left(x+2\right)}dx$.

Answer 1

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

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

$x=A(x-1)(x+2)+B(x+2)+C(x-1{)}^2$

Equating coefficients

$A+C=0$

$A+B-2C=1$

$-2A+2B+C=0$

On solving

$A=\frac{2}{9}C=\frac{-2}{9}B=\frac{1}{3}$

$\int \frac{x}{(x-1{)}^2(x+2)}dx=\frac{2}{9}\int \frac{1}{x-1}dx+\frac{1}{3}\int \frac{1}{(x-1{)}^2}dx-\frac{2}{9}\int \frac{1}{x+2}dx$

$I=\frac{2}{9}log|x-1|+\frac{1}{3}\frac{-1}{(x-1)}-\frac{2}{9}|x+2|+c$

$I=\frac{2}{9}log\frac{|x-1|}{|x+2|}-\frac{1}{3(x-1)}+C$