Answer
\[\frac{-3}{x+1}+\frac{5}{x+2}\]
Work Step by Step
The general form of the partial fraction decomposition is
\[\frac{2x − 1}{(x + 1)(x + 2)}=\frac{A}{x+1}+\frac{B}{x+2}\;\;\;...(*)\]
\[\Rightarrow 2x-1=A(x+2)+B(x+1)\]
\[\Rightarrow 2x-1=(A+B)x+(2A+B)\]
Comparing like coefficients
\[\Rightarrow A+B=2\;\;\;...(1)\]
\[\Rightarrow 2A+B=-1\;\;\;...(2)\]
Subtract equation (1) from equation (2)
\[\Rightarrow -A=3\Rightarrow A=-3\]
From (1)
\[\Rightarrow B=5\]
From (*)
\[\frac{2x − 1}{(x + 1)(x + 2)}=\frac{-3}{x+1}+\frac{5}{x+2}\]
Hence ,
\[\frac{2x − 1}{(x + 1)(x + 2)}=\frac{-3}{x+1}+\frac{5}{x+2}\;\;\;.\]
