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Home  >>  CBSE XII  >>  Math  >>  Integrals
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Integrate the function$\frac{\large 2x}{\large 1+x^2}$

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Toolbox:
  • $(i)\;Method \;of \;substitution:$
  • Given $\int f(x)dx$ can be transformed into another form by changing independent variable x to t by substituting x=g(t).
  • Consider $I=\int f(x)dx.$
  • Put x=g(t) so that $\frac{dx}{dt}=g'(t)$.
  • dx=g'(t)dt.
  • Thus $I=\int f(x)dx=\int f(g(t).g'(t))dt.$
$(ii)\;\int\frac{1}{x}dx=log x+c$.
 
Given $I=\int\frac{2x}{1+x^2}dx.$
 
Let $1+x^2=t.$
 
Differentiating on both sides we get,
 
2x dx=dt.
 
Substituing for $(1+x^2)$ and 2xdx we get,
 
$I=\int \frac{dt}{t}.$
 
Now integrating we get,
 
$I=log t+c$.
 
Now substituting back for t we get,
 
$log(1+x^2)+c$.
 
Hence $\int\frac{2x}{1+x^2}dx=log(1+x^2)+c.$

 

answered Jan 28, 2013 by sreemathi.v
 
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