Question

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

Answer 1

$$\begin{gathered} \mathrm{Let}I=\int \frac{\sin \sqrt{x}}{\sqrt{x}}dx \\ \mathrm{Let}\sqrt{x}=t \\ \Rightarrow \frac{1}{2\sqrt{x}}dx=dt \\ \Rightarrow \frac{dx}{\sqrt{x}}=2dt \\ \mathrm{Putting}\sqrt{x}=t\mathrm{and}\frac{dx}{\sqrt{x}}=2dt,\mathrm{we}\mathrm{get} \\ \therefore I=2\int \sin tdt \\ =-2\cos t+C\mathit{}\mathit{}\mathit{}\mathit{}\mathit{}\mathit{}\mathit{}\mathit{}\mathit{}\mathit{}\mathit{}\left(\mathit{\because }\mathit{}\mathit{t}\mathit{=}\sqrt{\mathit{x}}\right) \\ =-2\cos \sqrt{x}+C\mathit{}\mathit{}\mathit{}\mathit{} \end{gathered}$$