Antiderivatives of Products of Constants & Functions

Instructor: Laura Pennington

Laura has taught collegiate mathematics and holds a master's degree in pure mathematics.

Antiderivatives are an undoing of a derivative in a way. This lesson will review what an antiderivative is and will then go on to explain a particular rule that tells us how to find the antiderivative of the product of a constant and a function.

Antiderivatives

You are probably familiar with what the derivative of a function is as the slope of a function at a given point or as the rate at which the function's y-values are changing respect to the x-values at a given point. You also may be familiar with the various formulas and rules we can use to evaluate derivatives. What you may not be familiar with is that derivatives have an undoing of sorts, and that lies in the antiderivative.

Antiderivatives are used in many useful applications such as finding areas, volumes, and various points of a function. As their name implies, these antiderivatives undo derivatives. Basically, if f (x) is the derivative of the function F(x), then F(x) + C, where C is some constant, is the antiderivative of f (x). For example, the derivative of 2x is 2, so the antiderivative of 2 is 2x + C, where C is some constant.

The notation we use to indicate the antiderivative of a function f(x) is ∫ f(x) dx.


antidiffpro1


For instance, the derivative of x2 + 3x is 2x + 3. This tells us that the antiderivative of 2x + 3 is x2 + 3x + C, where C is some constant. Using our notation, we write the following:

  • ∫ 2x + 3 dx = x2 + 3x + C

Just as we have rules for finding various derivatives, we also have rules for finding various antiderivatives. This lesson is mainly concerned with how to find the antiderivative of the product of a constant and a function, so let's look at the rule we can use to do just this!

Antiderivative of the Product of a Constant and a Function

Suppose you've just found the antiderivative of the function f(x) = 5x4 to be F(x) = x5 + C. You go back to check your work, and realize that you were supposed to find the antiderivative of 2f(x), not f(x).

Well, shucks! Does that mean we have to start all over? Thankfully, no. Notice that 2f(x) is the product of the constant 2 and the function f(x). As it turns out, to find the antiderivative of the product of a constant and a function, we use the following rule:

  • cf(x) dx = cf(x) dx


antidiffpro2


That is, the antiderivative of a product of a constant and a function is equal to the constant times the antiderivative of the function.

Based on this rule, we can simply multiply the antiderivative of f(x) = 5x4, or F(x) = x5 + C, by 2 to correct our mistake!

∫ 2f(x) dx = 2 ∫ f(x) dx By the rule
2 ∫ f(x) dx = 2F(x) Because F(x) = ∫ f(x) dx
2F(x) = 2(x5 + C) Plugging in F(x) = x5 + C
2(x5 + C) = 2x5 + C This is our answer!

Ta-da! Problem solved, and we didn't have to start all over, we simply had to multiply the antiderivative we already found by 2, thanks to our rule! We have that the antiderivative of 2f(x) is 2x5 + C.

Notice, we still write C for the constant, and not 2C. This is because it is any constant, and multiplying it by 2 would just produce another constant, so we can still represent it as C, where C is any constant.

Basically, when we evaluate antiderivatives of a constant and a function using this rule, we don't need to worry about the constant C in our calculations, because it is just a placeholder for any constant. In fact, we can leave the C out of the calculations completely and then just add it back in when we get to our final answer. Let's do another example working with this rule!

Another Example

We have a nice rule for the derivative of the function sin(x), and that is that the derivative of sin(x) is cos(x). This tells us that the antiderivative of cos(x) is sin(x) + C. Based on this, could you figure out the antiderivative of 7cos(x)?

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