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The Fundamental Theorem of Calculus

The Fundamental Theorem of Calculus

At a Glance - Why the Choice of Antiderivative Doesn't Matter

Some say po-TAY-to, some say po-TAH-to. Some say to-MAY-to, some say to-MAH-to. Some choose to not include a constant when finding an antiderivative. Some do include the constant. Here's why it doesn't matter what we do with the constant.

Sample Problem

Find .

The simplest antiderivative of 4x3 is x4. Using the FTC with that antiderivative, we get

Now let's try the FTC with a different antiderivative. How about x4 + 3?

Notice how the extra "+ 3"s canceled each other out and we got 15 again. If we used some other antiderivative of 4x3, the same sort of thing would happen.

The moral of the story is that when evaluating a definite integral with the FTC, no matter which antiderivative you use, you should get the same answer every time. Since it doesn't matter which antiderivative you use, you may as well use the simplest one.

Exercise 1

Find  using the FTC with antiderivative x4 – π.

Exercise 2

Find  using the FTC and

(a) the antiderivative -cos t

(b) the antiderivative -cos t + 1

(c) the antiderivative -cos te

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