Antiderivatives: Think “undoing” derivatives Since: We say is the “antiderivative of

Definition: A function F is an antiderivative of f on an interval I if F’(x) = f(x) for x in I. Here is another antiderivative for Why?

Notice that for any constant, C, this is an antiderivative: C is called the ‘constant of integration’ And the function, F, above is the ‘general antiderivative’ of f

Here is the notation for finding an antiderivative (or indefinite integral) Integrand variable of integration constant antiderivative

Integration Rules: First, see that integration and differentiation are ‘inverses’ of each other. One undoes the other, like multiplication and division undo each other. ‘undoing’ a derivative ‘undoing’ an integral

Here is what I call the ‘Reverse Power Rule” Now, go meditate and hum: Increase the exponent by one and divide by the new exponent

Here is an example: Like derivatives, as long as we are adding and subtracting terms we can separate them and do them one at a time. to check, take the derivative, it should equal the integrand

Here is one more: There is a division, and there is not a ‘reverse’ quotient rule. We will change to a form we can handle, that is, x n undo adding fractions

Simplifying the exponents Now it is the form of adding x n terms and we can do the Reverse Power Rule on each term

A quick quiz for you: Remember, you can check your work by taking derivatives