In this lesson, you will learn the definition of antiderivative, the formula for the antiderivatives of powers of *x* and the formulas for the antiderivatives of trigonometric functions

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Definition of Antiderivative

A function *F* is called an antiderivative of *f* on an interval *I* if *F’*(*x*) = *f*(*x*) for all *x* in *I*.

The general antiderivative of *f*(*x*) = *x ^{n}* is

where *c* is an arbitrary constant.

* Example: *

Find the most general derivative of the function *f*(*x*) = *x*^{–3}

** Solution: **

Function |
Particular antiderivative |

cos |
sin x |

sin |
– cos |

sec2 |
tan |

sec |
sec |

** Example: **

Find antiderivative of the function

** Solution: **

Rewrite the given function as follows:

An introduction to indefinite integration of polynomials.

Basic Antiderivative Examples

This video has a few examples of finding indefinite integrals of trig functions

Definition of Antiderivatives

Antiderivatives are the opposite of derivatives. An antiderivative is a function that reverses what the derivative does. One function has many antiderivatives, but they all take the form of a function plus an arbitrary constant. Antiderivatives are a key part of indefinite integrals.

Antidifferentiation

Finding the antiderivatives of a function require a little backwards thinking. Since the derivative of the wanted antiderivative is the given function, checking for correctness is easy. You just take the derivative, and see if it is the given function. Also, antiderivatives of functions happen to be not just one function, but a whole family of functions. This family can be written as a polynomial plus c, where c stands for any constant.

You can use the Mathway widget below to practice Algebra or other math topics. Try the given examples, or type in your own problem. Then click "Answer" to check your answer.

*(Clicking "View Steps" on the answer screen will take you to the Mathway site, where you can register for a free ten-day trial of the software.)*

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