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How to Antidifferentiate Sums

Instructor: Laura Pennington

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

In this lesson, we'll review antiderivatives then explore how to antidifferentiate sums of functions using a formula for the antiderivative of a sum. We'll apply the formula to multiple examples to help solidify our understanding of the concept.

Antiderivatives

Suppose you are a rocket scientist, and you're working on a problem in which you just found the derivative of a function, F(x), to be f(x) = 8x7.

However, moments after you found the derivative, you lost the paper that had the original function on it. Uh-oh! How are you going to figure out what the original function is if you only know its derivative?

Not to worry! The answer lies in antiderivatives. The antiderivative of a function, f(x), is equal to the function F(x), such that f(x) is the derivative of F(x). The notation we use to indicate the antiderivative of f(x) is as follows:

  • Antiderivative of f(x) = ∫ f(x) dx

The antiderivative of a function, f, is the function that f is the derivative of. That is, if f(x) is the derivative of F(x), then we have the following:

  • f(x) dx = F(x) + C, where C is any constant


antidiffsum1


Therefore, the original function you were working with, F(x), is just equal to the antiderivative of 8x7, or the function whose derivative is 8x7. This function is F(x) = x8 + C, where C is some constant, because the derivative of x8 + C is 8x7.

  • ∫ 8x7 dx = x8 + C

You use some of your other calculations to determine that in this case C = 0, so you have your original function of F(x) = x8. Phew! That was close! On to the next task in solving your rocket scientist problem that you're working on!

How to Antidifferentiate Sums

The next task in this problem is to calculate the antiderivative of the sum of the function, f(x) = 8x7 and the function g(x) = sin(x). Thankfully, we have a nice rule that enables us to find the antiderivative of a sum of functions quite easily!

When we want to find the antiderivative of a sum of functions, f(x) + g(x), we use the following rule:

  • f(x) + g(x) dx = ∫ f(x) dx + ∫ g(x) dx

We see that to find the antiderivative of a sum of functions f(x) and g(x), we simply find the sum of the antiderivatives of those functions.


antidiffsum2


This is great news! We can use this to complete the next task in the problem you're working on. To find the antiderivative of the sum of f(x) = 8x7 and g(x) = sin(x), we find the sum of the antiderivatives of these two functions.

  • ∫ 8x7 + sin(x) dx = ∫ 8x7 dx + ∫ sin(x) dx

We already know the antiderivative of 8x7 is x8 + C, and since sin(x) is the derivative of -cos(x), we have that the antiderivative of sin(x) is -cos(x) + C. We plug these into our formula, and we have the antiderivative of the sum. We can leave the constant C out until the end of our calculations, because it is any constant. Adding two constants together just gives another constant, so we don't have to worry about including the general constant, C, in the calculations.

  • ∫ 8x7 + sin(x) dx = ∫ 8x7 dx + ∫ sin(x) dx = x8 + (-cos(x)) + C = x8 - cos(x) + C

We did it! We found the antiderivative of the sum of the functions, and our formula made it fairly easy to do! Let's consider another example.

Example

For a little more practice, consider the following facts:

  • 9 is the derivative of 9x.
  • 1/x is the derivative of ln(x).

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