Question

Integrate the rational functions.
$\int \frac{x}{(x+1)(x+2)}dx.$

Answer 1

$\int \frac{x}{(x+1)(x+2)}dx.Let\frac{x}{(x+1)(x+2)}=\frac{A}{(x+1)}+\frac{B}{(x+2)}\Rightarrow \frac{x}{(x+1)(x+2)}=\frac{A(x+2)+B(x+1)}{(x+1)(x+2)}\Rightarrow x=Ax+2A+Bx+B\Rightarrow x=x(A+B)+2A+B$
On equating the coefficients of x and constant term on both sides, we get
A+B=1 ......(i)
and 2A+B =0 .......(ii)
On subtracting Eq. (ii)From Eq. (i)we get $-A=1\Rightarrow A=-1$
Now, on putting the value of A in Eq.(i), we get -1+B =1 $\Rightarrow B=2$
$\therefore \int \frac{x}{(x+1)(x+2)}dx=\int \frac{(-1)}{x+1}dx+\int \frac{2}{x+2}dx=-log(x+1)+2log(x+2)+C=-log(x+1)+log(x+2{)}^2+C=log\left|\frac{(x+2{)}^2}{x+1}\right|+C[\because logb-loga=log\frac{b}{a}]$