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Question
Evaluate:\[\int\frac{\cos \sqrt{x}}{\sqrt{x}} \text{ dx }\]
Solution
\[\text{ Let I } = \int\frac{\cos\sqrt{x}}{\sqrt{x}} dx\]
\[\text{ Putting} \sqrt{x} = t\]
\[ \Rightarrow \frac{1}{2\sqrt{x}}dx = dt\]
\[ \Rightarrow \frac{dx}{\sqrt{x}} = 2 \text{ dt}\]
\[ \therefore I = 2\int\text{ cos t dt}\]
\[ = 2 \sin t + C ,\text{ where t }= \sqrt{x}\]
\[ = 2 \sin \sqrt{x} + C\]
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