Question

Evaluate:\[\int\frac{\cos \sqrt{x}}{\sqrt{x}} \text{ dx }\]

Answer 1

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