What is an inductor? Wikipedia, or HowStuffWorks
The following equation represents the voltage across the terminals of an inductor as a function of the current of the inductor:
The previous equations can be used to derive an equation which expresses the current as a function of the voltage.
Both sides of the equation are multiplied by the differential time dt. The rate at which i varies with t multiplied by a differential change in time equals a differential change in i (the two differential changes in time on the right side cancel each other out):
v dt = L di
Both sides of the equation are integrated. Choosing x and τ as the variables of integration respectively, i and t become limits on the integrals.
i(t) becomes the current corresponding to t and i(t0) becomes the value of the inductor current when inegration is initiated at t0
Typically the value of t0 is zero,