Answer:
Option B
Explanation:
$\frac{x+5}{(x+1)^{2}(x+2)}=\frac{A}{(x+1)^{}}+\frac{B}{(x+1)^{2}}+\frac{C}{(x+2)^{}}$ ..........(i)
$\Rightarrow$ $x+5= A(x+1)(x+2)+B(x+2)+c(x+1)^{2}$
$\Rightarrow$ $x+5= x^{2}(A+C)+x(3A+B+2C)+2A+2B+C$
on comparing the coefficients of like power of x , we get
$A+C=0$,
$3A+B+2C=1$ and $2A+2B+C=5$
on substituting A=-C in last two equations, we get
B-C=1 .....(ii)
and 2B-C=5 .......(iii)
On subtracting Eqs(ii) and (iii) , we get
$B=4 \Rightarrow C=3$ [from Eq.(ii)]
and hence A=-3
Now, $\frac{x+5}{(x+1)^{2}(x+2)}=\frac{-3}{(x+1)^{}}+\frac{4}{(x+1)^{2}}+\frac{3}{(x+2)^{}}$
On differentiating both sides w.r.t x, we get
$\frac{d}{dx}\left(\frac{x+5}{(x+1)^{2}(x+2)}\right)=$ $\frac{3}{(x+1)^{2}}-\frac{8}{(x+1)^{3}}-\frac{3}{(x+2)^{2}}$
