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