Answer
$$\int\sqrt t\sin(2t^{3/2})dt=-\frac{1}{3}\cos(2t^{3/2})+C$$
Work Step by Step
$$A=\int\sqrt t\sin(2t^{3/2})dt$$
We set $a=2t^{3/2}$, which means $$da=2\times\frac{3}{2}t^{1/2}dt=3\sqrt tdt$$ $$\sqrt tdt=\frac{1}{3}da$$
Therefore, $$A=\frac{1}{3}\int\sin ada$$ $$A=-\frac{1}{3}\cos a+C$$ $$A=-\frac{1}{3}\cos(2t^{3/2})+C$$
