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Home  >>  CBSE XII  >>  Math  >>  Integrals
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Integrate the function\[\frac{\cos\sqrt x}{\sqrt x}\]

$\begin{array}{1 1}2\sin \sqrt x+c \\ 2\cos \sqrt x+c \\ 2\sqrt \sin x+c \\2\sqrt \cos x+c \end{array}$

Can you answer this question?
 
 

1 Answer

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Toolbox:
  • (i)Method of substitution :
  • Given 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 $\frac{dx}{dt}=g'(t).$
  • $\Rightarrow $dx=g'(t)dt.
  • Thus $I=\int f(g(t).g'(t))dt.$
  • (ii)$\int \cos xdx=\sin x+c.$
Given $I=\int \frac{\cos {\sqrt x}}{\sqrt x}dx.$
 
Put $\sqrt x=t.$
 
$\frac{1}{2\sqrt x}dx=2dt.$
 
Now substituting for $\sqrt x$ and dx we get,
 
$I=2\int\cos tdt.$
 
On integrating we get,
 
$2\sin t+c$.
 
Substituting back for t we get,
 
$2\sin \sqrt x+c.$
 
Hence $\int \frac{\cos {\sqrt x}}{\sqrt x}dx=2\sin \sqrt x+c$.
 
 
 

 

answered Jan 29, 2013 by sreemathi.v
 
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