I've been directed here from the Electronic engineering website, since it's more of a math question. I need to demonstrate that given in the title, given this formula:
I know I have to replace i(t) with sin(t). But what interval should I choose for this integral? Should it be [0,1] as this is the interval that sin is defined in?
What you can show using that formula is that the average value of a sinusoidal function over its period is zero. Assuming that you know the basics of integral calculus, and taking into account that $t_1$ in that formula is arbitrary (so can be chosen equal to zero) you can simply evaluate this term:
$$\int_0^Ti(t)dt = \int_0^{2\pi}sin(t)dt = 0$$
where T is the period of a sinusoidal function.
So, dividing this result by $2\pi$, that is the period you still get zero.
Hope this helps.
D.