1. The Natural Logarithmic Function
Differentiation
Integration

2. Properties of the Natural Log Function
If a and b are positive numbers and n is rational, then the following properties are true:

3. The Algebra of Logarithmic Expressions

4. The Derivative of the Natural Logarithmic Function
Let u be a differentiable function of x

5. Differentiation of Logarithmic Functions

6. Differentiation of Logarithmic Functions

7. Differentiation of Logarithmic Functions

8. Differentiation of Logarithmic Functions

9. Differentiation of Logarithmic Functions

10. Logarithmic Properties as Aids to Differentiation
Differentiate:

11. Logarithmic Properties as Aids to Differentiation
Differentiate:

12. Logarithmic Differentiation
Differentiate:
This can get messy with the quotient or product and chain rules. So we will use ln rules to help simplify this and apply implicit differentiation and then we solve for y’…

13. Derivative Involving Absolute Value
Recall that the ln function is undefined for negative numbers, so we often see expressions of the form ln|u|. So the following theorem states that we can differentiate functions of the form
y= ln|u| as if the absolute value symbol is not even there.
If u is a differentiable function such that u≠0 then:

14. Derivative Involving Absolute Value
Differentiate:

15. Finding Relative Extrema
Locate the relative extrema of 𝑦=ln⁡( 𝑥 2 +2𝑥+3)
Differentiate:
𝑢= 𝑥 2 +2𝑥+3, 𝑢 ′ =2𝑥+2
𝑑𝑦 𝑑𝑥 = 2𝑥+2 𝑥 2 +2𝑥+3
Set = 0 to find critical points
2𝑥+2 𝑥 2 +2𝑥+3 =0
2x+2=0
X=-1, Plug back into original to find y
y=ln(1-2+3)=ln2 So, relative extrema is at (-1, ln2)

16. Homework
5.1 Natural Logarithmic Functions and the Number e Derivative #19-35,47-65, 71,79,93-96

17. General Power Rule for Integration
𝑥 𝑛 𝑑𝑥= 𝑥 𝑛+1 𝑛+1 +𝑐, 𝑛≠−1
Recall that it has an important disclaimer- it doesn’t apply when n = -1. So we can not integrate functions such as f(x)=1/x.
So we use the Second FTC to DEFINE such a function.

18. Integration Formulas
Let u be a differentiable function of x

19. Using the Log Rule for Integration

20. Using the Log Rule with a Change of Variables
1 4𝑥−1 𝑑𝑥
Let u=4x-1, so du=4dx and dx= 𝑑𝑢 4
1 𝑢 𝑑𝑢 4 = 𝑢 𝑑𝑢

21. Finding Area with the Log Rule
Find the area of the region bounded by the graph of y, the x-axis and the line x=3.

22. Recognizing Quotient Forms of the Log Rule
𝑥+1 𝑥 2 +2𝑥 𝑑𝑥→𝑢= 𝑥 2 +2𝑥, 𝑑𝑢=2𝑥+2𝑑𝑥
→ (𝑥+1) 𝑥 2 +2𝑥 𝑑𝑥= 1 2 𝑙𝑛 𝑥 2 +2𝑥 +𝑐 =𝑙𝑛 𝑥 2 +2𝑥 𝑐

23. Definition The natural logarithmic function is defined by
The domain of the natural logarithmic function is the set of all positive real numbers

24. u-Substitution and the Log Rule

25. Long Division With Integrals

26. How you know it’s long Division
If it is top heavy that means it is long division.
Example

27. Example 1

28. Continue Example 1

29. Example 2

30. Continue Example 2

31. Using Long Division Before Integrating

32. Using a Trigonometric Identity

33. Guidelines for integration
Learn a basic list of integration formulas. (including those given in this section, you now have 12 formulas: the Power Rule, the Log Rule, and ten trigonometric rules. By the end of section 5.7 , this list will have expanded to 20 basic rules)
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.
If you cannot find a u-substitution that works, try altering the integrand. You might try a trigonometric identity, multiplication and division by the same quantity, or addition and subtraction of the same quantity. Be creative.
If you have access to computer software that will find antiderivatives symbolically, use it.

34. Integrals of the Six Basic Trigonometric Functions

35. Homework
5.2 Log Rule for Integration and Integrals for Trig Functions (substitution)
#1-39, 47-53,  67