Should I use superposition if there is a sine wave current source and a DC voltage source in a circuit?
Start off with a DC analysis (Open circuit the current source, and the caps, inductors are short circuits), assuming that resistor in series with the voltage source is 2 ohms (Difficult to read), then with both capacitors open circuit and the inductor short circuiting the 8 ohm resistor you get 12V across 4 + 2 ohms = 6 ohms = 2A for the DC terms of I(t). 

Now, being careful of the fact that 2 radians/s is not the same thing as 2Hz, compute the AC impedances at the frequency of the current source and having replaced the DC voltage source with a short circuit solve for the AC component of I(t), then superposition gets you the answer.