So you might have been wondering, what makes an improper integral, well improper?
Sometimes people mistakenly think that integrals are only improper when either the upper limit
is positive infinity, or when the lower limit is negative infinity, or when both of those
happen at the same time and the interval is negative infinity to positive infinity. But
actually, even integrals that look totally normal, with real numbers for both the upper
and lower limit, something like this, even these integrals can sometimes be improper.
So what is it about an integral that makes
it improper, compared to just a "normal" definite integral? Well technically, an integral
is "improper" if at least one of the following is true. Number one, the interval
of integration is infinite, or number two, the integrand is discontinuous in the interval.
Let's talk about these two cases one at a time.
Number one. An integral is improper if the
interval of integration is infinite. This is the obvious case that we talked about before.
The interval of integration is made up of the limits of integration, so if the lower
limit is negative infinity, and/or if the upper limit of integration is positive infinity,
then the interval of integration is infinite. In that case, then automatically the integral
is considered improper. Luckily for us, these kinds of improper integrals are easy to identify,
because we always have the limits of integration right in front of us, so in this case we'll
know right away when we look at the integral if it's going to be improper.
Number two. An integral is improper if the
integrand is discontinuous in the interval. This one is harder. After all, what do we
even mean when we use the word "integrand"? Well, the integrand is just the function that
we're integrating. It's the value we always find between the integral symbol and
the dx that we put at the end. This is the function we're trying to integrate, and
if it has any kind of discontinuity inside the interval, then the integral in general
is considered to be improper. Even if neither of the limits of integration are infinite,
the integral is still improper because of the discontinuity in the function.
Now it's important to point out that the
discontinuity in the function has to be inside the interval of integration. So if we're
integrating from zero to four, and there's a discontinuity at x equals two, then there's
a discontinuity inside the interval. The function could be discontinuous all over the place
outside the interval, either to the left of x equals zero, or to the right of x equals
four. But none of those discontinuities would make the integral improper, because right
now we only care about the function between zero and four. Remember that the function
is also technically discontinuous inside the interval if it has a discontinuity at one
of the endpoints of the interval, either exactly at the left edge, or exactly at the right
edge.
As an example, let's think about the function one over x minus 2, which we can see has a
vertical asymptote, and therefore a discontinuity, at x equals 2. At first glance, all of these
integrals seem like totally normal definite integrals. But let's check to see if any
of them are improper. With this integrand, if we integrate over negative two to zero,
then the integral isn't improper, because there's no discontinuity on that interval.
If we integrate on zero to two, then the integral is improper, because there's a discontinuity
at the right edge of the interval. If we integrate on two to four, then the integral is improper,
because there's a discontinuity at the left edge of the interval. And if we integrate
on zero to four, then the integral is improper because there's a discontinuity inside
the interval between the left and right edge.
So now we know what defines an improper integral. Either the interval of integration is infinite
at the lower limit, the upper limit, or both, or the integrand is discontinuous somewhere
inside the interval of integration, possibly including at the edges of the interval.
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