Question

Integrate the function $\frac{x^2+x+1}{(x+1{)}^2(x+2)}$

Answer 1

Let $\frac{x^2+x+1}{(x+1{)}^2(x+2)}=\frac{A}{(x+1)}+\frac{B}{(x+1{)}^2}+\frac{C}{(x+2)}$ ............. (1)
$\Rightarrow x^2+x+1=A(x+1)(x+2)+B(x+2)+C(x^2+2x+1)$
$\Rightarrow x^2+x+1=A(x^2+3x+2)+B(x+2)+C(x^2+2x+1)$
$\Rightarrow x^2+x+1=(A+C)x^2+(3A+B+2C)x+(2A+2B+C)$
Equating the coefficients of $x^2,x$,and constant term, we obtain
$A+C=1$
$3A+B+2C=1$
$2A+2B+C=1$
On solving these equations, we obtain
$A=2,B=1$, and $C=3$
From equation (1), we obtain
$\frac{x^2+x+1}{(x+1{)}^2(x+2)}=\frac{-2}{(x+1)}+\frac{3}{(x+2)}+\frac{1}{(x+1{)}^2}$
$\int \frac{x^2+x+1}{(x+1{)}^2(x+2)}dx=-2\int \frac{1}{x+1}dx+3\int \frac{1}{(x+2)}dx+\int \frac{1}{(x+1{)}^2}dx$
$=-2\log |x+1|+3\log |x+2|-\frac{1}{(x+1)}+C$