A technician studies a circuit that has a rectifier in it…

A technician studies a circuit that has a rectifier in it. A rectifier lets the current pass in one direction only, so the charge keeps growing. The current at time $t$ seconds is $i(t) = \sin^3 t\cos^2 t$ amperes. The charge $q$ coulombs that has passed a point in the wire satisfies $\dfrac{dq}{dt} = i$, and at $t = 0$ no charge has passed, so $q(0) = 0\ \text{C}$. She notices that the power of $\sin t$ is odd, so she keeps one factor $\sin t$ for $du$ and writes the rest with $\cos t$. With the substitution $u = \cos t$ she finds $q(t)$ in the form $-\dfrac{\cos^3 t}{3} + \dfrac{\cos^5 t}{5} + C$. Find the constant $C$, which is fixed by the condition at $t = 0$. Then find the charge that has passed at $t = \dfrac{\pi}{2}\ \text{s}$, and at $t = \dfrac{\pi}{3}\ \text{s}$. Give $C$ in coulombs and each charge in coulombs, to four decimal places.