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If $\frac{d}{d x}\left(\frac{2 x+1}{(x+1)^2(x-2)}\right)=\frac{\mathrm{A}}{(x-2)^2}+\frac{\mathrm{B}}{(x+2)^3}+\frac{\mathrm{C}}{(x+1)^2}$ then $\mathrm{A}+\mathrm{B}+\mathrm{C}=$
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Verified Answer
The correct answer is:
$\frac{-2}{3}$
$\begin{aligned} & \text { now, } \log y=\log (2 x+1)-2 \log (x+1)-\log (x-2) \\ & \Rightarrow \frac{1}{y} \frac{d y}{d x}=\frac{2}{2 x+1}-\frac{2}{x+1}-\frac{1}{x-2} \\ & \text { or } \frac{d y}{d x}=\frac{y}{2 x+1}-\frac{2 y}{x+1}-\frac{y}{x-2}\end{aligned}$
from (i)
$\frac{d y}{d x}=\frac{2(x+1)(x-2)-2(x-2)(2 x+1)-(2 x+1)(x+1)}{(x+1)^3(x-2)^2}$
On comparing with R.H.S
We get
$\mathrm{A}=\frac{-5}{9}, \mathrm{~B}=0$ and $\mathrm{C}=\frac{-1}{9}$
now $\mathrm{A}+\mathrm{B}+\mathrm{C}=\frac{-6}{9}=\frac{-2}{3}$
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