1. Natural Logarithms
Integration

2. Log Rule for Integration

3. Using Log Rule

4. Using Log Rule

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

6. Finding Area with the Log Rule

7. Recognizing Log Rule
Remember that the integral of the natural logarithm occurs only when the function in the denominator is raised to the 1st power.

8. Examples
p. 325
On-line examples

9. Guidelines for Integration
1. Learn a basic list of integration formulas
2. Find an integration formula that resembles all or part of the integrand, and, by trial and error, find a coice of u that will make the integrand conform to the formula

10. Guidelines for Integration
3. If you can’t 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.
4. If you have access to computer software that will find antiderivatives symbolically, use it.

11. Integrals of Trigonometric Functions

12. Average Price The demand equation for a product is
Find the average price p on the interval 40 ≤ x ≤ 50.