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Integrate the function$\frac{\large 2x}{\large 1+x^2}$

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  • $(i)\;Method \;of \;substitution:$
  • 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,
Hence $\int\frac{2x}{1+x^2}dx=log(1+x^2)+c.$


answered Jan 28, 2013 by sreemathi.v