Question

The value of \[\int\frac{\cos \sqrt{x}}{\sqrt{x}} dx\] is

Options

• 2 cos \[\sqrt{x}\]

• \[\sqrt{\frac{\cos x}{x}} + C\]

• sin \[\sqrt{x} + C\]

• 2 sin \[\sqrt{x} + C\]

Answer 1

2 sin \[\sqrt{x} + C\]

 

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