Mathematics

# 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 $B=2$
$\therefore\displaystyle \frac {x}{(x+1)(x+2)}dx=\frac {-1}{(x+1)}+\frac {2}{(x+2)}$
$\Rightarrow\displaystyle \int \frac {x}{(x+1)(x+2)}dx=\int \frac {-1}{(x+1)}+\frac {2}{(x+2)}dx$
$=-\log|x+1|+2\log |x+2|+C$
$=\log (x+2)^2-\log |x+1|+C$
$\displaystyle =\log \frac {(x+2)^2}{(x+1)}+C$

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Subjective Medium Published on 17th 09, 2020
Questions 203525
Subjects 9
Chapters 126
Enrolled Students 105

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