Question

\[\int\frac{\sin\sqrt{x}}{\sqrt{x}} dx\]

Answer 1

\[\int\frac{\sin \sqrt{x}}{\sqrt{x}}dx\]
\[\text{Let} \sqrt{x} = t\]
\[ \Rightarrow \frac{1}{2\sqrt{x}} = \frac{dt}{dx}\]
\[ \Rightarrow \frac{dx}{\sqrt{x}} = 2dt\]
\[Now, \int\frac{\sin \sqrt{x}}{\sqrt{x}}dx\]
\[ = 2\int\text{sin t dt}\]
\[ = 2 \left[ - \cos t \right] + C\]
\[ = - 2 \cos \sqrt{x} + C\]