A coil is connected to a circuit in a radio receiver, and the current in it dies away while it…

A coil is connected to a circuit in a radio receiver, and the current in it dies away while it oscillates. A meter records the current, and the engineer wants the total charge. At $t$ seconds the current is $i(t) = e^{-2t}\sin 3t$ amperes. The charge $q$ in coulombs that has passed a point satisfies $\dfrac{dq}{dt} = i$, and $q(0) = 0\ \text{C}$. An engineer wants $q(t)$. She sees that parts must be used twice, and that the original integral comes back. So she calls the integral $I$, applies parts two times, and solves an equation for $I$. She finds $q(t) = -\dfrac{e^{-2t}}{13}\left(2\sin 3t + 3\cos 3t\right) + K$. Find the constant $K$ from $q(0) = 0$. Then find the charge that has passed after $1$ second and after $2$ seconds. Last, find the total charge that passes as $t$ becomes very large, since $e^{-2t}$ then tends to zero. Give $K$ and the charges in coulombs to four decimal places.