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Home  >>  CBSE XII  >>  Math  >>  Integrals
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Integrate the rational functions\[\frac{2x}{x^2+3x+2}\]

$\begin{array}{1 1} 4\log|x+2|-2\log|x+1|+c \\ 4\log|x+2|+2\log|x+1|+c \\ 2\log|x+2|-\log|x+1|+c \\2\log|x+2|+\log|x+1|+c \end{array} $

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1 Answer

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Toolbox:
  • $(i)\;$Form of the rational function\[\frac{px+q}{(x+a)(x+b)}\]
  • $\;$Form of the partial function\[\frac{A}{x+a}+\frac{B}{x+b}\]
  • $(ii)\;\int\frac{dx}{(x+a)}=log|x+a|+c.$
Given:$I=\int\frac{2x}{x^2+3x+2}dx=\int\frac{2x}{(x+2)(x+1)}dx.$
 
Let $\frac{2x}{(x+2)(x+1)}=\frac{A}{x+2}+\frac{B}{x+1}.$
 
$\Rightarrow 2x=A(x+1)+B(x+2)$
 
equating the coefficients of x,
 
2=A+B----(1)
 
Equating the constant terms,
 
0=A+2B------(2)
 
On solving
 
A+B=2----(1)
 
A+2B=0-----(2)
 
Subtracting equ(1) and equ(2)
 
-B=2$\Rightarrow B=-2.$
 
Substituting for B in equ(1) we get,
 
A=4;Hence A=4 and B=-2
 
I=$\int\frac{4.dx}{(x+2)}+\int\frac{(-2)dx}{(x+1)}$
 
$\;\;\;=4\int\frac{dx}{x+2}-2\int\frac{dx}{x+1}$.
 
On integrating we get,
 
$\;\;\;=4log|x+2|-2log|x+1|+c.$

 

answered Feb 5, 2013 by sreemathi.v
 
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