Antiderivatives of Constants & Power Functions

Instructor: David Karsner
In the study of calculus, you are frequently called on to find the antiderivative of a function. In this lesson, you'll learn how to find antiderivatives of constants and power functions.

What Is an Antiderivative?

Imagine you are hiking through the woods and come across an animal track left in the mud. Can you determine what animal created that track? This process of working backwards to find out what created the animal track is similar to the the process of working backwards to find the antiderivative of a function.

In calculus, you have to find the derivative of many different types of functions. The derivative is the rate of change of a function at a given point, x. For example, the derivative of f(x)=x2-4x+2 would be 2x-4.

Finding the antiderivative involves starting with a function and then finding what other function would have created the first function by taking the derivative. If the function was f(x)=2x-4, an antiderivative would be F(x)=x2-4x+2. The antiderivative and the integral have similar definitions. A definite integral, which is the area under a curve from point a to b, is a value, and the antiderivative is a function. The indefinite integral and the antiderivative are even closer in definition, and for the purpose of this lesson, can be used as synonyms. Derivatives are denoted with f(x), whereas the antiderivative of f(x) is denoted with a capital letter F(x).

The Antiderivative of a Power Function

Power functions have the form of f(x)=axb. They have one term in which you have a constant (a) times a variable x raised to some power. Examples of power functions would f(x)=3x2 and g(x)=.5x5. To find the antiderivative of a power function follow these steps.

  1. Notice the exponent and add one.
  2. Divide the term by the exponent plus one.
  3. The result of the second step plus a constant C will be the antiderivative.

The antiderivative will have the +C at the end of it. You'll learn more about this later in the lesson. The antiderivative of a power function will always have a degree, the largest exponent in the function, that is one greater than the original function.

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