Integral Calculus, The Indefinite Integral

The Indefinite Integral

The indefinite integral on the half open interval [a,b) is the limit of the integral over [a,x] as x approaches b. Clearly this is a one-sided limit. Note that b might be +∞. We can also compute the integral from -∞ to b, or from -∞ to +∞, i.e. the integral over the entire x axis.

If f is a complex or vector function, with the standard topology, we know a sequence of multi-dimensional values approaches a limit in n space iff the components approach the corresponding limits. Thus the indefinite integral of a complex function f over the interval [a,b) exists iff the real and imaginary components have indefinite integrals on the same open interval.

We may become a bit loose with our terminology. Consider the function 1 over the square root of x from 0 to 1. This is unbounded, hence it is not technically integrable. Yet we will often say it is integrable, with area 2. We really mean the indefinite integral exists, and is 2. (We will prove this later.) When functions are unbounded, or the domain of integration is infinite, "integral" usually means "indefinite integral", and integrable means the indefinite integral exists.