Determine the indefinite integral

$$\int e^{2t}\sin (3t)dt$$

A robotic underwater probe tracks its velocity along a direct path. Its velocity, $v$ ${ms}^{-1}$, at time $t$ seconds is modeled by the equation

$$v(t)=e^{2t}\sin (3t),t\ge 0$$

The probe starts at a fixed marker at $t=0$ and travels until it first comes to a momentary halt at time $T>0$.

Show that the total distance the probe has travelled from the marker is exactly

$$\frac{3}{13}\left(e^{\frac{2\pi }{3}}+1\right)metres$$