Prove the identity

$$\frac{\sin 2x}{1+{\tan }^{2}x}\equiv 2\sin x{\cos }^{3}x$$

The rate of airflow $R$ into a mechanical respirator, in litres per second, is modeled by the function

$$R(t)=\frac{10\sin 6t}{1+{\tan }^{2}3t}$$

where $t$ is the time in seconds. Find an expression for the total volume of air $V=\int R(t)dt$.