Question

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

Answer 1

\[\text{ Let I } = \int\frac{\sin\sqrt{x}}{\sqrt{x}} dx\]
\[\text{ Let }\sqrt{x} = t\]
\[ \Rightarrow \frac{1}{2\sqrt{x}}dx = dt\]
\[ \Rightarrow \frac{dx}{\sqrt{x}} = 2 dt\]
\[\text{ Putting }\sqrt{x} = t  \text{ and }\frac{dx}{\sqrt{x}} = 2 \text{ dt  , we get} \]
\[ \therefore I = 2\int\text{ sin  t  dt}\]
\[ = - 2 \text{ cos  t} + C \left( \because t = \sqrt{x} \right)\]
\[ = - 2 \cos \sqrt{x} + C\]