1. 5 Logarithmic, Exponential, and Other Transcendental Functions
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2. Calculus Warm-Up

3. Calculus Warm-Up 1/27/11

4. 5.7 Inverse Trigonometric Functions: Integration
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5. Objectives
Integrate functions whose antiderivatives involve inverse trigonometric functions.
Use the method of completing the square to integrate a function.
Review the basic integration rules involving elementary functions.

6. Integrals Involving Inverse Trigonometric Functions
The derivatives of the six inverse trigonometric functions fall into three pairs. In each pair, the derivative of one function is the negative of the other.
For example,
and

7. Integrals Involving Inverse Trigonometric Functions
When listing the antiderivative that corresponds to each of the inverse trigonometric functions, you need to use only one member from each pair.
It is conventional to use arcsin x as the antiderivative of rather than –arccos x.

8. Inverse Trigonometric
Function Integrals:

9. Integrals Involving Inverse Trigonometric Functions

10. Example

11. Example
Let u = 3x
du = 3 dx

12. Ex. Integrate by substitution.
Let u = ex
du = ex dx

13. Let u = 4 – x2 du = -2x dx Final Answer
Ex. Rewriting the integrand as the sum of two quotients.
Let u = 4 – x2
du = -2x dx
Final Answer

14. Do long division and then rewrite the
Ex. Integrating an improper rational function.
Do long division and then rewrite the
integrand as the sum of two quotients.

15. Ex:
Let u = x – 2
du = dx

16. Ex. Completing the square when the
leading coefficient is not 1.
First, factor out a 1/2
Now complete the square
in the denominator.
Let u = x – 2
du = dx

17. Find the area of the region bounded by the graph of
f(x) = , the x-axis, and and 2 1

18. Factor out a neg. inside the rad.

19. Adding and Subtracting Common Denominators
Ex: Find the integral:
The derivative of x2 + 2x + 2 is 2x + 2, so to get it, add
and subtract

20. Adding and Subtracting Common Denominators
Now, combine the first two integrals.

21. Review of Basic Integration Rules
You have now completed the introduction of the basic integration rules. To be efficient at applying these rules, you should have practiced enough so that each rule is committed to memory.

22. Example 6 – Comparing Integration Problems
Find the following integrals using the formulas and techniques you have studied so far.

23. Example 6 – Solution
a. You can find this integral (it fits the Arcsecant Rule).

24. Example 6 – Solution
b. You can find this integral (it fits the Power Rule).

25. Example 6 – Solution
c. You cannot find this integral using the techniques you have studied so far.

26. Review of Basic Integration Rules

27. In Class Assignment (now):
pg. 385: , 53, 63

28. Homework Day 1
Pg odds + 53, 63
Homework 5.7 Day 2
MMM pgs. 207, 208

29. HWQ 1/27
Evaluate the integral: