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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?
• Well, essentially.

In application, you'll have additional constraints, which will narrow down the possibilities.
• Are their any good websites that provide integration problems?
• 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!
• Okay after studying so far I understand that:
derivatives are used to find the minimum or maximum of something to optimize something.
• 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?
• 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!
• 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?
• Yes, that is correct. That will be a useful understanding when you are solving differential equations, which will depend heavily on those arbitrary constant.
• Will there be videos about solving rational,irrational and hyperbolic function integrals?
• What is the range of points that you need an area for?
• 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.
• 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
• is this calculus 2 material?
• Calc 1 should include at the very least a brief lesson on this. Calc 2 goes much farther in-depth with integrals.
• 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?
• 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!