In words, equation [1] states that y at timet is equal to the integral of x() from minus infinity up to time t. Now, recall thederivative property of the Fourier Transform for a function g(t):

Let's rewrite this Fourier property:

We can substitute h(t)=dg(t)/dt [i.e. h(t) is the time derivative of g(t)] into equation [3]:

Since g(t) is an arbitrary function, h(t) is as well and equation [4] gives a general result:

Equation [5] is "mostly" true. The reason it lacks completeness is that we used equation [3] to derive it - and note that the derivative of g(t) removes any constant: that is, the derivative of g(t) is equal to the derivative of the function g(t)+b, for any constant b. Because of this oversite, equation [5] is almost correct. If the total integral of g(t) is 0, then equation [5] directly follows from equation [3], and we have:

If the total integral of g(t) is not zero, then there exists some constantc such that the total integral of g(t)-c is zero:

That is,c is the "average value" of the function g(t), which is also often called the "dc term" or the "constant term". Using some math and theFourier Transform of the impulse function, we have the general formula for the Fourier Transform of the integral of a function:

The Dirac-Delta impulse function in [7] isexplained here.

The integration property is used and the constant in [8] is utilized on the Fourier Transform pagefor the unit step function, which should help clear things up if the above is not clear.