Definite integral, Indefinite integral, Area under the curve of the function, Primitive function, Newton-Leibniz theorem, Fundamental theorem of calculus.
It is time to get familiar with computing integrals. There are two types: definite and indefinite integrals.
Definite integration deals with calculating the area under the curve of a function.
Here is a function
and the area under its curve from a to b is:
Indefinite integration works a totally different way.
The reason we call it indefinite is that it does not have a and b limits for the integration, we just simply integrate the function:
The indefinite integral of f(x) is a function that is called the primitive function.
The symbol for the primitive function is F(x), and its main property is that if we differentiate it, we get f(x).
So, this indefinite integration is nothing else than the reverse of differentiation.
This is why it is sometimes called antidifferentiation.
Let's see a few examples.
Here is this one, for instance:
We need a function whose derivative is 2x.
There is such a function, namely
Here comes another one:
There is also a function, whose derivative is
If we can recall,
If one knows what absolute value is, it won't be a very disturbing piece of news that we need to use that here, too. The reason is that we want to integrate
for negative x values, too.
But lnx only likes positive x values. Luckily, the absolute value solves this minor problem.
However, it is enough to remember this:
Finally, let's see one more:
What should we differentiate to get x2?
This is almost perfect; we just have to divide it by 3.
And just one more thing. If we differentiate x2, of course it will be 2x, but
This means any constant can be added to x2.
And here too.
And now, let's see the relationship between definite and indefinite integrals.
The theorem that describes this relationship is one of the most important theorems in the entire history of mathematics.
It was developed at the end of the 1600s by two men at the same time: an English physicist named Newton and a German philosopher named Leibniz.
If f(x) is integrable on the closed interval [a, b], and it has a primitive function on this interval, then
This thing here means the change of the primitive function, so first we substitute b, then we substitute a, and then subtract it.
Let's try how this theorem works: let's find the area under the curve of x2 from 0 to 1.
Here comes the primitive function, and we have to find its change from 0 to 1.
If we can't remember the primitive function, then we are in trouble.
Let's compute the area under the curve of
from 0 to 1.
It isn’t particularly challenging to write the formula of what we should integrate.
However, we have no idea what the primitive function is, and that is a showstopper.
So, the crux of the matter is finding the primitive function, in other words, finding the indefinite integral.
We can’t push it off anymore: it is time to master that skill.