If $$x(t) = x_0 u_0 t - \frac{1}{2}gt^2$$, then $$d^2x/dt^2$$ is equal to

The integral $$\displaystyle{\int_0^x 2 \sin{2t} \;dt}$$ is equal to

The integral $$\displaystyle{\int_0^\infty x e^{-x} \;dx}$$ is equal to

The $$\displaystyle{\lim_{\mu \to 0} \frac{\sin{\mu t}}{\mu}}$$ is equal to

The $$\displaystyle{\lim_{\omega \to \omega_0} \frac{\cos{\omega t} - \cos{\omega_0 t}}{\omega_0^2 - \omega^2}}$$ is equal to