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### Course: AP®︎/College Calculus AB > Unit 6

Lesson 7: The fundamental theorem of calculus and definite integrals# Antiderivatives and indefinite integrals

What's the opposite of a derivative? It's something called the "indefinite integral". Created by Sal Khan.

## Want to join the conversation?

- So, is it all about writing "possible functions" from its derivatives?(27 votes)
- Well, essentially.

In application, you'll have additional constraints, which will narrow down the possibilities.(29 votes)

- Are their any good websites that provide integration problems?(5 votes)
- go to http://www.math.ucdavis.edu/~kouba/ProblemsList.html

and scroll down to beginning integral calculus.

Hope this helped =).(35 votes)

- How can we use derivatives, integrals and anti-derivatives in real life situations or in an occupation like engineering? I'm kinda confused here, if anyone can give an example and a detailed explanation I would really appreciate it. I really want to understand this intuitively. Thanks in advance!(16 votes)
- Okay after studying so far I understand that:

derivatives are used to find the minimum or maximum of something to optimize something.(5 votes)

- I'm currently in Grade 10, and our math subjects are mainly on trigonometry, quadratic equations, and general algebra 2 stuff. I can safely say that I've learnt all Grade 10, 11 and even 12 math level subjects with Khan Academy since I am so fascinated by math. I was wondering whether I should start learning calculus since it's so profound and deep. Should I just wait it out till college or can I start learning this by myself?(9 votes)
- Start now!

Start here at the beginning of the Pre Calculus Track: https://www.khanacademy.org/math/precalculus

Do the work in the order presented - if you don't get one or two things, ask a question.

If you are not getting a lot of things, best do some appropriate review.

Have Fun!(8 votes)

- cant the anti derivative of 2x be x^2+c1+c2+...+cn ?

is it so that all the constants are summed and they become one constant?

Which gives us just one constant notation c?

Have I understood correctly?(4 votes)- Yes, that is correct. That will be a useful understanding when you are solving differential equations, which will depend heavily on those arbitrary constant.(6 votes)

- Will there be videos about solving rational,irrational and hyperbolic function integrals?(12 votes)
- What is the range of points that you need an area for?(0 votes)

- In the integral notation, why is there a dx at the end? I've looked everywhere for an explanation, but I just can't find one. I would really appreciate an answer.(3 votes)
- The symbol dx has different interpretations depending on the theory being used. In Leibniz's notation, dx is interpreted as an infinitesimal change in x and his integration notation is the most common one in use today. If the underlying theory of integration is not important, dx can be seen as strictly a notation indicating that x is a dummy variable of integration; if the integral is seen as a Riemann integral, dx indicates that the sum is over subintervals in the domain of x; in a Riemann–Stieltjes integral, it indicates the weight applied to a subinterval in the sum; in Lebesgue integration and its extensions, dx is a measure, a type of function which assigns sizes to sets; in non-standard analysis, it is an infinitesimal; and in the theory of differentiable manifolds, it is often a differential form, a quantity which assigns numbers to tangent vectors. Depending on the situation, the notation may vary slightly to capture the important features of the situation. For instance, when integrating a variable x with respect to a measure μ, the notation dμ(x) is sometimes used to emphasize the dependence on x. Source: http://en.wikipedia.org/wiki/Integral#Terminology_and_notation(8 votes)

- is this calculus 2 material?(3 votes)
- Calc 1 should include at the very least a brief lesson on this. Calc 2 goes much farther in-depth with integrals.(6 votes)

- If a Derivative shows the rate of change of a curve & if an Integral shows the area under the curve.

Then what is an Antiderivative? What is it used for?(4 votes)- At first, mathematicians studied three (or four if you count limits) areas of calculus. Those would be derivatives, definite integrals, and antiderivatives (now also called indefinite integrals).

When you learn about the fundamental theorem of calculus, you will learn that the antiderivative has a very, very important property. There is a reason why it is also called the indefinite integral.

I won't spoil it for you because it really is incredible!(4 votes)

- So what's the difference with indefinite integrals and definite integrals? Is that even a thing? How would you calculate it? Are they the same as antiderivatives? If not, what's the difference?(3 votes)
- A definite integral gives you the area under a function between two bounds a and b. The answer to a definite integral is a number.

An indefinite integral gives you the family of antiderivatives of a function. The answer to an indefinite integral is a function plus C.

Definite and indefinite integrals are connected by the Fundamental Theorem of Calculus. It says that to calculate a definite integral (the area between two bounds), you have to find the difference between the indefinite integral (antiderative) evaluated at the upper bound and lower bound.(5 votes)

## 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.