 A useful property of the definite integral, suppose f of x is continuous over some interval between a and b, then the definite integral over that interval is guaranteed to exist. This means that for most functions, and for most intervals, the definite integral will exist. So, this integral exists, and that means it's worth putting in the effort to find its value. Now, sometimes we might not be able to find an anti-derivative, for example this function, but because it's continuous over the interval, we know that the definite integral exists, and we can try to find it using some other method. And likewise for this function, it's continuous over the interval, so the definite integral exists, regardless of our ability to actually calculate it. And one of the strange features about being a mathematician is that once you know that a solution exists, actually finding it, you don't really care so much about. And in fact, the first question you ask is, well, is there something I can't find? And this leads us to the idea of improper integrals. And this gives us two problematic cases. First, what if the interval isn't finite, and second, what if the function isn't continuous? And this leads to two slightly different types of improper integrals. First, the interval of the integration might extend to plus or minus infinity, and we call this a type 1 improper integral. Or the integrand may be discontinuous at some point in the interval, and this would be a type 2 improper integral. So how can we extend our definition of the definite integral to include these cases? For a type 1 improper integral, where maybe our upper bound goes to infinity, we'll define it in terms of a limit as the upper bound goes to infinity of a definite integral. And similarly, if our lower bound is minus infinity, we'll define that as the limit of a proper integral. Now while the definite integrals exist, the improper integral is defined as a limit, and so there is a possibility the limit doesn't exist. And so if the limits exist, we say the integral converges to the limiting value, otherwise the integral diverges. For example, suppose we want to evaluate this integral, since the upper limit is infinity, this is a type 1 improper integral, not that it really matters, but what does matter is its value, if it exists, is going to be defined as a limit. And from our definition, that will be... So the last thing we have to do here is to take care of the limit, let's find the definite integral first. Then evaluate, and now we can take the limit. One. And so the limit is the value of the improper integral. The definition of type 1 improper integrals only allow one of the limits to be infinite, but what if they both are? For that, we'll fall back on the useful property of the definite integral. We can split the interval. So suppose the integral from minus infinity to C and from C to infinity both converge, then the improper integral from minus infinity to positive infinity will be the sum of the two convergent integrals. For example, consider this integral. Our definition says that if the integral from C to infinity and from minus infinity to C both exist, then their sum will be the integral from minus infinity to infinity. But what's C? To find out, we'll use a time-honored academic tradition, procrastination. We won't worry about C until we need to. So let's just find the integral from C to infinity, which will be the limit as B goes to infinity of the definite integral. We'll evaluate the definite integral. Find the limit, which exists regardless of the value of C. And so the improper integral has a definite value. Similarly, we can find the integral from minus infinity to C. First, we'll rewrite it as a limit. Evaluate the definite integral. Take the limit. And again, this will exist regardless of the value of C. Since both integrals converge, the integral from minus infinity to positive infinity is the sum of the two integrals. And it turns out we don't need to know the value of C.