In this video, I will explain what an improper integral is,
I will give the definition, and an example.
One type of improper integral is any integral of this form:
The integral from a number a to infinity.
Before we can even try to compute something like this,
we need to decide what does it even mean
to integrate a function all the way to infinity.
We need a definition, and I will motivate the definition with an example.
In this example, I want to compute the integral from 0 to infinity
of 1/(1+x²)dx.
Geometrically, it represents this area:
the area between the x-axis and this graph starting at x = 0
and then going all the way to infinity.
And the first thing to notice is that
I don't even know what this is supposed to mean yet.
For all I know, this doesn't make sense.
I know how to integrate a function between two numbers a and b,
and we know when that's guaranteed to be integrable or not,
but what does it mean to integrate between 0 and infinity?
My first task is to figure out what that means.
I'm going to try to appeal to your geometric intuition
to give a possible interpretation of it,
and after computing this example
I will define in general what it means to integrate between 0 and infinity.
As I was saying, while I can't integrate between 0 and infinity,
I can integrate between 0 and any other number b.
So if I stop at this point, I know how to compute this area.
And that would be the integral between 0 and b of the same function.
Then what I can do is, well, I cannot plug in b = ∞,
but I can then take b and make it bigger and bigger and bigger;
in other words, I can make b approach infinity.
Or what's the same, I can take the limit as b approaches infinity of this quantity.
Well, that's a limit I can try to compute.
I don't know whether it will exist, perhaps it does, perhaps it doesn't,
but at least this is something that makes sense geometrically,
and this is how I'm going to choose to define any integral like this.
I'm going to finish this computation,
and then I will write a general definition of
an integral between a number and infinity.
First I have to calculate the integral, then take the limit.
To compute this integral I find an antiderivative of 1/(1+x²),
that is arctan(x),
and then I have to evaluate at b and at 0.
So this is this limit.
We know the arctangent of 0 is 0,
and then arctangent has a limit when the variable goes to infinity,
which happens to be pi/2.
Okay, so I was not expecting that, but here's the answer.
The area between the axis and this curve, going all the way to infinity, is pi/2.
Now let's take what I did in this example and let's generalize it.
I'm going to use the same idea to define in general
what it means to write an integral between a number and infinity.
We are ready for a definition.
This type of integral is called in some books an improper integral of type 1.
I don't really care about the name.
The problem of this integral is that the domain goes all the way to infinity;
it's unbounded.
So how do we define it?
I have a real number a and I begin with a function f
that is continuous from a to infinity, a is included.
And then I want to define the integral of f from a to infinity.
We're going to define it like in the previous example.
First, for any real number b > a,
I can compute the integral from a to b (that will be a number),
and now I get a function that depends on b.
And then I take the limit as b approaches infinity.
Why is this a good definition?
Since f is continuous from a to infinity, it will also be continuous from a to b,
and therefore we know it's integrable.
So from a to b, the integral always exists for any b > a.
The second step, the limit, may or may not exist,
so we're going to say that this is the integral assuming the limit exists.
And that's why I require the function to be continuous.
It's the easiest way to guarantee that it is integrable from a to b for every b > a.
We will also say that this integral is convergent whenever the limit exists.
And when it doesn't, we say the integral is divergent,
which means simply the integral is not well-defined, it's not a number.
Converent means it is a number, divergent could mean that it is infinity,
or minus infinity, or any other reason why the limit does not exist.
Now that we have a definition, we can compute lots of examples.
In the next few videos, I will compute these four.
I invite you to try to compute them before watching the answer
in the corresponding videos.
The first three are similar to this, but the last one is different.
It is improper in a different way that I haven't explained yet,
and I have yet to define.
But the challenge is to try to figure out
what is wrong or different about that integral and how it should be defined
before watching the answer in the corresponding video.