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 = t²

$ \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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