Question

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

Answer 1

$\int \frac{xdx}{(x+1)(x+2)}$

Using partial fraction,

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

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

$x=A(x+2)+B(x+1)$

Put $x=-2,$ we get

$-2=A.O+B(-1)$

$B=2$

Put $x=-1$, we get

$-1=A\left(1\right)+B.O$

$A=-1$

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

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

$=-ln(x+1)+2ln(x+2)+c$

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

$=ln\left(\frac{(x+2{)}^2}{x+1}\right)+C$

$\therefore \int \frac{xdx}{(x+1)(x+2)}=ln\left(\frac{(x+2{)}^2}{x+1}\right)+C$