Integrate the function: x x - Vidyadip

Question

Integrate the function: $\frac{{\cos \sqrt x }}{{\sqrt x }}$

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Answer

Let $I = \int {\frac{{\cos \sqrt x }}{{\sqrt x }}dx} ...(i)$
Putting $\sqrt x = t$
$\Rightarrow x = t^2$
$ \Rightarrow \frac{{dx}}{{dt}} = 2t$
$ \Rightarrow $ dx = 2tdt
$\therefore$ From eq. $ (i),I = \int {\frac{{\cos t}}{t}2tdt} $
$ = 2\int {\cos t\,dt} $
$= 2 \sin t + c$
$ = 2\sin \sqrt x + c$

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