Determine the indefinite integral
∫e2tsin(3t) dt \int e^{2t} \sin(3t) \, dt ∫e2tsin(3t)dtA robotic underwater probe tracks its velocity along a direct path. Its velocity, vvv ms−1\text{ms}^{-1}ms−1, at time ttt seconds is modeled by the equation
v(t)=e2tsin(3t),t≥0 v(t) = e^{2t} \sin(3t), \quad t \geq 0 v(t)=e2tsin(3t),t≥0The probe starts at a fixed marker at t=0t = 0t=0 and travels until it first comes to a momentary halt at time T>0T > 0T>0.
Show that the total distance the probe has travelled from the marker is exactly
313(e2π3+1) metres \frac{3}{13}\left(e^{\frac{2\pi}{3}} + 1\right) \text{ metres} 133(e32π+1) metres
