Question

Evaluate: $\int \frac{\cos \sqrt{x}}{\sqrt{x}}dx$

Answer 1

$$\begin{gathered} \mathrm{Let}I=\int \frac{\cos \sqrt{x}}{\sqrt{x}}dx \\ \mathrm{Putting}\sqrt{x}=t \\ \Rightarrow \frac{1}{2\sqrt{x}}dx=dt \\ \Rightarrow \frac{dx}{\sqrt{x}}=2dt \\ \therefore I=2\int \cos tdt \\ =2\sin t+C\mathit{}\mathit{}\mathit{}\mathit{}\mathit{,}\mathit{}\mathrm{where}t\mathit{=}\sqrt{x} \\ =2\sin \sqrt{x}+C\mathit{}\mathit{}\mathit{}\mathit{}\mathit{} \end{gathered}$$