1. Aim: Integrating Natural Log Function Course: Calculus Do Now: Aim: How do we integrate the natural logarithmic function?

2. Aim: Integrating Natural Log Function Course: Calculus Log Rule for Integration Rules of Differentiation Rules of Integration Enables integration of rational functions

3. Aim: Integrating Natural Log Function Course: Calculus Model Problems Multiple & Divide by 4 u = 4x – 1u’ = 4 Substitute u Log rule Back substitute

4. Aim: Integrating Natural Log Function Course: Calculus Model Problem Alternate form of Log Rule Look for quotients in which numerator is the derivative of denominator. du = u’dx

5. Aim: Integrating Natural Log Function Course: Calculus Model Problem u = x 2 + 1u’ = 2x Look for quotients in which numerator is derivative of denominator.

6. Aim: Integrating Natural Log Function Course: Calculus Model Problems u = x 3 + x u’ = 3x 2 + 1 u = tanx u’ = sec 2 x u = x 2 + 2x u’ = 2x + 2 Look for quotients in which numerator is derivative of denominator.

7. Aim: Integrating Natural Log Function Course: Calculus Model Problems Look for quotients in which numerator is a degree higher or equal to denominator long division u = x 2 + 1 u’ = 2x

8. Aim: Integrating Natural Log Function Course: Calculus Do Now: Aim: How do we integrate the natural logarithmic function?

9. Aim: Integrating Natural Log Function Course: Calculus Model Problem – Change of Variables Look for quotients in which numerator is derivative of denominator. u = x + 1  x = u – 1 Substitute u Rewrite 2 fractions Rewrite 2 Integrals

10. Aim: Integrating Natural Log Function Course: Calculus Model Problem – Change of Variables Look for quotients in which numerator is derivative of denominator. Rewrite 2 Integrals Integrate Simplify Back- substitute

11. Aim: Integrating Natural Log Function Course: Calculus Guidelines for Integration 1. Memorize a basic list of integration formulas. (20) 2.Find an integration formula that resembles all or part of the integrand, and, by trial and error, find a choice of u that will make the integrand conform to the formula. 3.If you cannot find a u-substitution that works, try altering the integrand. You might try a trig identity, multiplication and division by the same quantity, or addition and subtraction of the same quantity. Be creative. 4.If you have access to computer software that will find antiderivatives symbolically, use it.

12. Aim: Integrating Natural Log Function Course: Calculus Model Problem u = x Will log rule apply? What does u equal? u = x lnx u = lnx u’ = 1/x Divide N & D by x Substitute u Log Rule Back- substitute

13. Aim: Integrating Natural Log Function Course: Calculus Integrals of Trig Functions u = cosx u’ = -sinx Substitute u Log Rule Back- substitute

14. Aim: Integrating Natural Log Function Course: Calculus Integrals of Trig Functions u = sec x + tan xu’ = sec x tanx + sec 2 x Back- substitute Log Rule

15. Aim: Integrating Natural Log Function Course: Calculus Integrals for Basic Trig Functions

16. Aim: Integrating Natural Log Function Course: Calculus Model Problem 1 + tan 2 x = sec 2 x - Pythagorean Identity

17. Aim: Integrating Natural Log Function Course: Calculus Model Problem The electromotive force E of a particular electrical circuit is given by E = 3sin2t, where E is measured in volts and t is measured in seconds. Find the average value of E as t ranges from 0 to 0.5 second. u = 2t du = 2dt  1.379 volts

18. Aim: Integrating Natural Log Function Course: Calculus Do Now: Aim: How do we integrate the natural logarithmic function?

19. Aim: Integrating Natural Log Function Course: Calculus Model Problem