Question

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

Answer 1

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

$\Rightarrow x=A(x+2)+B(x+1)\_\_\_\_\_\_\_\_\_\_1$

than

$\left[x=-2\right]$ putin eqn

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

$B=2$

after $x=-1$

$-1=A(-1+2)+0$

$A=-1$

than after

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

$\Rightarrow -\log (x+1)+2\log (x+2)+c$

$\Rightarrow 2\log (x+2)-\log (x+1)+c$

$\Rightarrow \log \left[\frac{{(x+2)}^2}{(x+1)}\right]+c$

$\Rightarrow \log \left[\frac{{(x+2)}^2}{(x+1)}\right]+c$