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Evaluate : $ \int\large\frac{\cos \sqrt x}{\sqrt x}$$dx $

1 Answer

  • 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 $I=\int f(g(t).g'(t))dt.$
Step 1:
$I=\int\large\frac{\cos\sqrt x}{\sqrt x}$$dx$
Put $\sqrt x=t$
$\Rightarrow \large\frac{1}{2\sqrt x}$$dx=dt$
$\large\frac{dx}{\sqrt x}$$=2dt$
Step 2:
Now substituting this we get,
$I=2\int \cos tdt$
$\;\;=2\sin t+C$
$\;\;=2\sin\sqrt x+C$
answered Nov 8, 2013 by sreemathi.v