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Integrate the rational functions
$\frac{2 x}{x^2+3 x+2}$
$\frac{2 x}{x^2+3 x+2}$
Solution:
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Verified Answer
Let $\frac{2 x}{x^2+3 x+2}=\frac{2 x}{(x+1)(x+2)}=\frac{\mathrm{A}}{\mathrm{x}+1}+\frac{\mathrm{B}}{\mathrm{x}+2}$
$\Rightarrow 2 x=\mathrm{A}(x+2)+\mathrm{B}(x+1) \quad \ldots(i)$
Putting $x=-1,-2$ in (i), we get : $A=-2 \& B=4$
$\begin{aligned}
&\therefore \int \frac{2 x}{x^2+3 x+2} d x=-2 \int \frac{d x}{x+1}+4 \int \frac{d x}{x+2} \\
&=-2 \log |x+1|+4 \log |x+2|+C
\end{aligned}$
$\Rightarrow 2 x=\mathrm{A}(x+2)+\mathrm{B}(x+1) \quad \ldots(i)$
Putting $x=-1,-2$ in (i), we get : $A=-2 \& B=4$
$\begin{aligned}
&\therefore \int \frac{2 x}{x^2+3 x+2} d x=-2 \int \frac{d x}{x+1}+4 \int \frac{d x}{x+2} \\
&=-2 \log |x+1|+4 \log |x+2|+C
\end{aligned}$
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