Integrate the functions in Exercises: x x - Vidyadip

Question

Integrate the functions in Exercises:
$\frac{\cos\sqrt{\text{x}}}{\sqrt{\text{x}}}$

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Answer

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

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