Question

\[\int\frac{e^\sqrt{x} \cos \left( e^\sqrt{x} \right)}{\sqrt{x}} dx\]

Answer 1

\[\int\frac{e^\sqrt{x} \cdot \cos \left( e^\sqrt{x} \right)}{\sqrt{x}}dx\]
\[\text{Let e}^\sqrt{x} = t\]
\[ \Rightarrow e^\sqrt{x} \times \frac{1}{2\sqrt{x}} = \frac{dt}{dx}\]
\[ \Rightarrow \frac{e^\sqrt{x}}{\sqrt{x}}dx = 2dt\]
\[Now, \int\frac{e^\sqrt{x} \cdot \cos \left( e^\sqrt{x} \right)}{\sqrt{x}}dx\]
\[ = 2\int\text{cos t dt} \]
\[ = 2 \sin t + C\]
\[ = 2 \sin \left( e^\sqrt{x} \right) + C\]