Question

Find the charge q(t) flowing through an element if the current is

i(t) = 3e

−t + 20 cos(15t − π

6

⁄ )uA

q(0) = 1uC

Answer 1

$i(t)= 3e^{-t}+20\:cos\left(15t-\frac{\pi }{6}\right)\\ q= \int i(t) dt\\ \int \:3e^{-t}+20\cos \left(15t-\frac{\pi }{6}\right)dt\\ \mathrm{Apply\:the\:Sum\:Rule}:\quad \int f\left(x\right)\pm g\left(x\right)dx=\int f\left(x\right)dx\pm \int g\left(x\right)dx\\ =\int \:3e^{-t}dt+\int \:20\cos \left(15t-\frac{\pi }{6}\right)dt\\ =-3e^{-t}+\frac{4}{3}\sin \left(15t-\frac{\pi }{6}\right)\\ =-3e^{-t}+\frac{4}{3}\sin \left(15t-\frac{\pi }{6}\right)+C\\ q(0) = 1uC \implies c=-\frac{4}{3}\\ q=-3e^{-t}+\frac{4}{3}\sin \left(15t-\frac{\pi }{6}\right)-\frac{4}{3}\\$