Antiderivative: Rules, Formula & Examples

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  • 0:00 Overview of Antiderivatives
  • 1:36 Antiderivative Formula
  • 3:56 Basic Rules of Antiderivatives
  • 6:15 Antiderivatives of…
  • 7:49 Lesson Summary
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Lesson Transcript
Instructor: Emily Cadic

Emily has a master's degree in engineering and currently teaches middle and high school science.

Learn about one of the foundations of calculus, the antiderivative, including the key guidelines for performing antiderivative calculations. You'll also have the chance to study a series of examples that walk you through typical problems found in introductory calculus.

Overview of Antiderivatives

There are multiple approaches to explaining the meaning of the term antiderivative, but the easiest one to understand is the graphical explanation. However, we cannot have a discussion of antiderivatives without acknowledging its partner in crime…you guessed it: the derivative.

The derivative is defined as the slope of the line running tangent to a function at a specific point. For example, the picture below shows the function y = x² in blue. A dotted tangent line has been drawn for three different points. You'll notice that the slope is doubling as you move along the x-axis; if the slope is twice the number you plug into the function as we see here, then the derivative is 2x.


On the other hand, the antiderivative is defined as the area underneath a function within a specific boundary. For example, if our function is y = 2x (the derivative from the previous example), the area underneath the graph is a triangle.


You can find the antiderivative at different points along the line using the formula for the area of a triangle, which is ½*base*height:


Can you see the function that is forming? It's y = x², the function we started with.

Antidifferentiation is the process of calculating the antiderivative of a function, just as differentiation is the process of calculating the derivative of a function. Let's go over the instructions for performing antidifferentiation calculations.

Antiderivative Formula

Now that we understand the physical basis for the antiderivative, it's time to reveal the formula that we'll use to calculate them. Using this formula to find the antiderivative of a function is fairly easy because you don't have to concern yourself with what its graph looks like. And for non-trigonometric functions, it's really quite simple:


You may be seeing some of this notation for the first time. In mathematics, f(x) is just a generic way of indicating a function. The a is used as a placeholder for any constant number, and the n stands for an exponent. The term C is also a constant, but it serves a different purpose that will be explained later in the lesson. Finally, we use F(x) to represent the antiderivative of function f(x).

Example One

It's time for our first 'whoa - I just did calculus' moment. Let's utilize this new formula in an example problem.

Example One: Given the function f(x) = 3x² , find F(x).

Solution: Remember, asking for F(x) is the same as asking us to find the antiderivative of the function f(x).

Step One: Identify the parts of the original function: constant a = 3, n = 2.

Step Two: Substitute the values from step one into the antiderivative formula:


Step Three: Report your final answer: f(x) = x³ + C.

Step Four : Check your answer by taking the derivative of x³.


Basic Rules of Antiderivatives

Let's expand on what we have just learned by going over some additional guidelines you'll need to solve antiderivative problems.

Rule One: Not all constants are treated the same in antidifferentiation problems.

  • The antiderivative of a standalone constant is a is equal to ax.
  • A multiplier constant, such as a in ax, is multiplied by the antiderivative as it was in the original function. For example, if f(x) = ax, F(x) = ½*a*x².

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