Finding a Fourier series of a Burst Fired sine wave having consecutive 2 cycles on & 2 cycles off?
Since your waveform repeats itself after 4 cycles of the sine wave, your overall time period is $$T = 8*\pi/\omega$$

Coming to solving the integrals, Since there are two parts when you look at a single period of the repeating wave, you break the integral into two parts,
$$\int_{t_0}^{t_0+T}i(t)dt = \int_{t_o}^{t_0+T/2}i(t)dt + \int_{t_o + T/2}^{t_0+T}i(t)dt$$

In the first half cycle, You have a sine wave, $$\int_{t_o}^{t_0+T/2}i(t)dt = \int_{t_o}^{t_0+T/2}sin(\omega*t)dt$$
In the second half cycle, There is no wave, $$\int_{t_o + T/2}^{t_0+T}i(t)dt = 0$$

I believe you can solve the rest