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Integrate the function $x\;\sqrt{1+2x^2}$

1 Answer

  • 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}{dx}=g'(t).
  • dx=g'(t)dt.
  • Thus $ I=\int f(g(t).g'(t))dt.$
Given $ I=x\sqrt {1+2x^2}dx.$
Let $t=1+2x^2 $.
On differentiating we get,$4xdx=dt.$
$\Rightarrow \;xdx=\frac{dt}{4}.$
Substituting for t and dt we get,
$I=\int \sqrt tdt.$
On integrating we get,
Substituting back for t we get,
Hence $ \int x\sqrt {1+2x^2}dx=\frac{2}{3}(1+2x^2)^\frac{3}{2}+c.$


answered Jan 28, 2013 by sreemathi.v