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6 Integration Antiderivatives and the Rules of Integration

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Clicker Question 1 Are you here? – A. Yes – B. No – C. Not sure

Clicker Question 2 What is the derivative function of f (t) = sin(3t 2 + 5) ? – A. cos(3t 2 + 5) – B. -cos(3t 2 + 5) – C. 6t cos(3t 2 + 5) – D. -6t cos(3t 2 + 5) – E. cos(6t)

Clicker Question 3 What is the most general antiderivative of f (x) = x – x -2 ? – A. 5x 4 + 2x -3 + C – B. 1/5 x 6 + 4x + ½ x -1 + C – C. 1/6 x 6 + 4x + x -1 + C – D. 1/6 x 6 + 4x - x -1 + C – E. I’m clueless

Clicker Question 4 What is the most general antiderivative of g(x) = (x + 2)(x + 3) ? – A. (1/2 x 2 + 2x )(1/2 x 2 + 3x ) + C – B. 1/3 x 3 + 6x + C – C. 1 – D. 1/3 x 3 + 5/2 x 2 + 6x + C – E. 2x + 5

The Definite Integral (9/6/13) What does it mean to “integrate a function” over some part of its domain? That is, given a function f (x) defined on an interval [a, b], what does mean? How can we compute this number?

The Fundamental Theorem Given a function f(x) on an interval [a,b], the Fundamental Theorem of Calculus tells us how the definite integral (a number) and the antiderivative (a function) are related. Part 1 says you can get an antiderivative of f by turning the definite integral into a function of the right-hand endpoint. (This part is less used – more theoretical.) Part 2 says that we can compute the definite integral provided we can find an antiderivative. Then we just evaluate that at the endpoints and subtract. (Used!)

FTC - A Quick Outline Given a function f (x ) on [a, b ]: F (x ) an antiderivative of f Part 1:  (by freeing up right endpoint) Part 2:  (by evaluating F (b) – F (a ) )

Examples Example of Part 1: Q: What is an antiderivative of cos(x 2 )? A: Example of Part 2 (you’ve seen lots!): Q: What is ? A: arctan(6) – arctan(2)

Clicker Question 5 Evaluate – A. 37 – B. ln(37) – ln(5) – C. ½(ln(1 + x 2 )) – D. ½(ln(37) – ln(5)) – E. 32

About Finding Anti-Derivatives We are assuming that you know the basic antiderivative facts and that you are familiar with the antiderivative techniques of algebraic manipulation and substitution. We will develop more techniques shortly. Remember that antiderivative only have two rules: Constant Multiplier and Sum and Difference.

Assignment for Monday Review as needed antiderivatives, definite integrals, and the Fundamental Theorem of Calculus. On Monday we will stop reviewing and move forward! Have a good weekend (but not too good!).