See the video below for a thorough explanation of what antiderivatives are, where they come from, and what you’re in store for in the following lessons. Note that there are trig functions referenced in the video briefly. If you don’t really care about trig, you can can ignore what is said about the sines and cosines.

Antiderivatives and Indefinite Integrals

An antiderivative of a function $f$ is any function $F$ such that
$$F'(x)=f(x)$$
wherever $f$ is defined.

The indefinite integral (or just integral, when the context is clear) of a function $f$ is the collection of all antiderivatives of $f$. The indefinite integral of $f$ is denoted as
$$\int f(x)\ dx.$$

Furthermore, if $F$ is any antiderivative of $f$, then
$$\int f(x)\ dx=F(x)+C$$
where $C$ represents an unknown constant.

You can think of antiderivatives as “backwards derivatives” because, in order to find antiderivatives and indefinite integrals, we will be “un-deriving” functions using rules similar to what we learned for finding derivatives. It is worth mentioning that most of the functions you have seen have derivatives, but there are some that don’t have antiderivatives that can be written in terms of functions we are used to dealing with.

Lesson Content

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The Power Rule for Antiderivatives

Initial Value Problems: Solving for C

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