CHAPTER 07.07: IMPROPER INTEGRATION: Background
In this segment, we're going to talk about some background about improper integrals. So before we talk about what numerical methods to use to do improper integration, let's talk about what an improper integral means. So if you look at a typical integral, it's given in the form a to b, f of x dx, a definite integral, and what improper integral is that there are . . . there can be several cases where you will consider a particular integral to be improper. One is where a or/and b are infinite. That means they can be, a can be plus infinity, a can be minus infinity, b can be minus infinity, and b can be plus infinity, so any of these combinations, or you don't necessarily have to have each, a also having an infinite and b also having an infinite limit, one of them can be. So if a or/and b are infinite, then it's considered to be an improper integral. Of course, we are only talking about integrals in which, where the integral is valid, in terms of that it's not unbounded. And the second one is where the integrand . . . where the integrand is singular. So you may have an integrand where, like integrand meaning f of x, is singular at some point, either at the endpoints, or at an endpoint, or somewhere between the two limits of integration. So the best thing to do is to look at some examples, that might be the best thing to do. So the ones where your lower or upper limit are infinite, you could have something like 1 to infinity, e to the power minus x squared, dx, let's suppose, that's an improper integral. You could have minus infinity to -2, e to the power minus x squared, dx, for example, that's also an improper integral. You could have 1 divided by square root of x minus 1, dx, going from 1 to, let's suppose 10, that's an improper integral, based on the fact that the integrand here, at x equal to 1, which is the lower limit, it is becoming, the denominator is becoming 0, so 1 divided by 0 is infinite. You could . . . you could also, let's suppose somebody gives you something like this, says 1 to 10, 1 divided by x minus 1, dx. Now, this is an improper integral also, but it's not defined, because what's going to happen is that when you do this integration, that will not turn out to be a bounded number, or a finite number. So we are only talking about improper integrals for which the integral does exist, because that's the one where you can find the value numerically, hopefully. Now, you can also have integrals like this, you can have f of x dx, going from -5 to +5, where f of x is equal to 1 divided by square root of x, for x greater than 0, and it's equal to 1 divided by square root of minus x, for x less than 0. So you could have integrals where the infinite nature, or x . . . yeah, x less or equal to 0, or greater than or equal to 0, so where the integrand becomes infinite, not necessarily at the endpoints, but somewhere in the . . . somewhere . . . at some point between the lower limit and upper limit of integration. Sometimes people think that in order for a . . . for an integral to be improper, that only at endpoints it can have infinite values, or singular values, but that's not necessarily true, as you can see for this particular integral. So these are the kind of examples which are existing for improper integrals. In the next segment, we will talk about how we can integrate such improper integrals. And that is the end of this segment. |