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Home  >>  CBSE XII  >>  Math  >>  Integrals

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

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  • 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 {x+2}.$
Let t=x+2 $\Rightarrow x=t-2.$
Hence dt=dx.
Substituting for t and dt we get,
$I=\int (t-2)\sqrt tdt.$
$\;\;\;=\int(t\sqrt t-2\sqrt t)dt.$
$\;\;\;=\int t^\frac{3}{2}dt-2\int t^\frac{1}{2}dt.$
On integrating we get,
Substituting back for t we get,


answered Jan 28, 2013 by sreemathi.v
edited Jan 28, 2013 by sreemathi.v