- Topics At a Glance
- Integrating the Velocity Function
- Negative Velocity
- Change in Position
- Total Distance Travelled vs. Change in Position
- About the Fundamental Theorem of Calculus (FTC)
**Using the FTC to Evaluate Integrals**- Integrating with Letters
- Order of Limits of Integration
**Why the Choice of Antiderivative Doesn't Matter**- Average Values
- Units
- Word Problems
- The Second Fundamental Theorem of Calculus
- Letters
- Antiderivatives
- Finding Derivatives
- In the Real World
- I Like Abstract Stuff; Why Should I Care?
- How to Solve a Math Problem

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.

Find .

The simplest antiderivative of 4*x*^{3} is *x*^{4}. Using the FTC with that antiderivative, we get

Now let's try the FTC with a different antiderivative. How about *x*^{4} + 3?

Notice how the extra "+ 3"s canceled each other out and we got 15 again. If we used some other antiderivative of 4*x*^{3}, 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 *x*^{4} – π.

Exercise 2

Find using the FTC and

(a) the antiderivative -cos *t*

(b) the antiderivative -cos *t* + 1

(c) the antiderivative -cos *t* – *e*