\(I=\int \sin (\sqrt{k}) d k\) on \(k \in(0, \infty)\)
Put \(k=t^2 \Rightarrow d k=2 t d t\)
\(\begin{aligned}
\therefore \quad I & =2 \int t \sin t d t \\
& =-2 t \cos t-2 \int 1.(-\cos t) d t \quad \text{(Integration by parts)}
\end{aligned}\)
\(\begin{aligned}
& =-2 t \cos t+2 \sin t+c \\
& =2[\sin (\sqrt{k})-\sqrt{k} \cos (\sqrt{k})]+c
\end{aligned}\)