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## The fundamental theorem of calculus and definite integrals

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# Antiderivatives and indefinite integrals

## Video transcript

We know how to take
derivatives of functions. If I apply the derivative
operator to x squared, I get 2x. Now, if I also apply
the derivative operator to x squared plus
1, I also get 2x. If I apply the derivative
operator to x squared plus pi, I also get 2x. The derivative of
x squared is 2x. Derivative, with respect to x
of pi of a constant, is just 0. Derivative, with
respect to x of 1, is just a constant, is just 0. So once again, this is just
going to be equal to 2x. In general, the
derivative, with respect to x of x squared
plus any constant, is going to be equal to 2x. The derivative of x squared,
with respect to x, is 2x. Derivative of a constant,
with respect to x, a constant does not
change with respect to x, so it's just equal to 0. So you have-- You apply
the derivative operator to any of these
expressions and you get 2x. Now, let's go the
other way around. Let's think about
the antiderivative. And one way to think
about it is we're doing the opposite of
the derivative operator. The derivative operator,
you get an expression and you find it's derivative. Now, what we want to do,
is given some expression, we want to find what it
could be the derivative of. So if someone were
to tell-- or give you 2x-- if someone were to
say 2x-- let me write this. So if someone were to ask you
what is 2x the derivative of? They're essentially asking
you for the antiderivative. And so you could say, well, 2x
is the derivative of x squared. But you could also say 2x is the
derivative of x squared plus 1. You could also say that 2x is
the derivative of x squared plus pi, I think you
get the general idea. So if you wanted to write it
in the most general sense, you would write that
2x is the derivative of x squared plus some constant. So this is what
you would consider the antiderivative of 2x. Now, that's all
nice, but this is kind of clumsy to have to write
a sentence like this, so let's come up with some notation
for the antiderivative. And the convention
here is to use kind of a strange
looking notation, is to use a big elongated s
looking thing like that, and a dx around the
function that you're trying to take the
antiderivative of. So in this case, it would
look something like this. This is just saying this is
equal to the antiderivative of 2x, and the antiderivative
of 2x, we have already seen, is x squared plus c. Now, you might be
saying, why do we use this type of crazy notation. It'll become more obvious when
we study the definite integral and areas under curves and
taking sums of rectangles in order to approximate
the area of the curve. Here, it really
should just be viewed as a notation for
antiderivative. And this notation right over
here, this whole expression, is called the
indefinite integral of 2x, which is another
way of just saying the antiderivative of 2x.