Answer
$$\displaystyle\displaystyle\int{\sin \sqrt{at}dt}=\frac{2}{a}\sin \sqrt{at}-\frac{2}{a}\sqrt{at}\cos \sqrt{at}+C
$$
Work Step by Step
$
I=\displaystyle\int{\sin \sqrt{at}dt}$
$\displaystyle{\left[\begin{matrix}
u=\sqrt{a}\sqrt{t}& dt=\frac{2\sqrt{t}du}{\sqrt{a}}& \sqrt{t}=\frac{u}{\sqrt{a}}\\
\end{matrix}\right]}
$
$I=\displaystyle\int{\sin u\frac{2\sqrt{t}du}{\sqrt{a}}}\\
I=\displaystyle\int{\sin u\frac{2}{\sqrt{a}}\frac{u}{\sqrt{a}}du}\\
I=\displaystyle\frac{2}{a}\int{u\sin udu}$
$\displaystyle{\left[\begin{matrix}
u=u& du=1\\
dv=\sin u& v=-\cos u\\
\end{matrix}\right]}$
$I=\displaystyle\frac{2}{a}\left( -u\cos u-\int{-\cos udu} \right)\\
I=\displaystyle\frac{2}{a}\left( -u\cos u+\int{\cos udu} \right)\\
I=\displaystyle\frac{2}{a}\left( -u\cos u+\sin u \right) +C\\
I=\displaystyle\frac{2}{a}\left( \sin u-u\cos u \right) +C\\
I=\displaystyle\frac{2}{a}\sin u-\frac{2}{a}u\cos u+C\\
I=\displaystyle\frac{2}{a}\sin \sqrt{at}-\frac{2}{a}\sqrt{at}\cos \sqrt{at}+C
$
