Do Now: 1. Find the derivative of the following functions a) b) Agenda: I. Do Now II. Indefinite integral III. Reverse power rule IV. Integral rules V. Homework Take out: pencil, notebook, calculator Homework: Handout Objectives: You will be able to find anti-derivatives and indefinite integrals using the reverse power rule.

 Since you are all such experts at derivatives, its time to move on.  The next HUGE topic we’ll be covering is integrals.  Today, we’ll take our first step into integrals, with anti-derivatives  Essentially, an anti derivative, is a derivative in reverse

 An antiderivative of the function f(x), is another function F(x), whose derivative is f(x).  In other words, an anti-derivative of f(x) is a function F(x), where F’(x) is f(x).  THAT SOUNDS CONFUSING, but really its not, an anti-derivative is just a derivative in reverse.  The process of finding the antiderivative is called antidifferentiation.

 The reverse power rule is a tool for taking anti-derivatives of polynomials.  If you have a function:  Then the anti-derivative, F(x), is

 Take the reverse power rule slowly at first.  Raise the exponent by 1 first!  Divide by that new exponent Check!

 Find the derivative of the functions below: Same! Two different functions, same derivative!

All 3 of these curves have the same derivative! Thus, they are ALL anti- derivatives of some function. Their slopes are the same at each x value! This comes from the fact that the derivative of a constant =0

 An indefinite integral represents the entire set of anti-derivatives (there are an infinite number of them).  The indefinite integral of f(x) is F(x)+C, We call C the constant of integration, and it could be anything! We’d need more information to figure out what it is.

 We use the integral symbol to represent the anti-derivative. Integral Symbol The integrand We’ll learn about this later. Let’s you know you are integrating with respect to x Anti- Derivative Constant of Integration

 Roots and Denominators!  Dividing by a fraction is the same as multiplying by its…  Reciprocal!

 Integrate the following:

 All of the common derivatives you have already memorized can be reversed to figure out common integrals.

 Memorize ‘em all! You should know them already.

 Before we start working with derivatives, we need to learn a few properties of integrals to make our lives easier. Integral of a constant: EX Slap an x on it!

 Integral of a constant times a function:  EX You can pull out constants +C or – C….it don’t matter!

Addition and Subtraction Ex The integral of a sum is the sum of the integrals!

Multiplication! NOOOOO! We need reverse product and quotient rules!