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Question

Integrate the rational function   $$\cfrac {x}{(x+1)(x+2)}$$


Solution

Let $$\displaystyle \frac {x}{(x+1)(x+2)}=\frac {A}{(x+1)}+\frac {B}{(x+2)}$$
$$\Rightarrow x=A(x+2)+B(x+1)$$
Equating the coefficients of $$x$$ and constant term, we obtain
$$A+B=1,2A+B=0$$
On solving, we obtain
$$A=-1$$ and $$B=2$$
$$\therefore\displaystyle  \frac {x}{(x+1)(x+2)}dx=\frac {-1}{(x+1)}+\frac {2}{(x+2)}$$
$$\Rightarrow\displaystyle  \int \frac {x}{(x+1)(x+2)}dx=\int \frac {-1}{(x+1)}+\frac {2}{(x+2)}dx$$
$$=-\log|x+1|+2\log |x+2|+C$$
$$=\log (x+2)^2-\log |x+1|+C$$
$$\displaystyle =\log \frac {(x+2)^2}{(x+1)}+C$$

Mathematics
RS Agarwal
Standard XII

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