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5.9 lesson filled in.notebook February 21, 2014
5
Copyright © Cengage Learning. All rights reserved.
Logarithmic, Exponential, and Other Transcendental Functions
Inverse Trigonometric Functions: Integration
Copyright © Cengage Learning. All rights reserved.
5.9
5.9 lesson filled in.notebook February 21, 2014
• 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.
Objectives
Integrals Involving Inverse Trigonometric Functions
5.9 lesson filled in.notebook February 21, 2014
Find the derivative:
So what is the integral of ?4
5.9 lesson filled in.notebook February 21, 2014
Integrals Involving Inverse Trigonometric Functions
Why is there only 3 formulas instead of 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
5.9 lesson filled in.notebook February 21, 2014
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.
*So we don't need these since they are just the negatives of the other three.
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.
Integrals Involving Inverse Trigonometric Functions
5.9 lesson filled in.notebook February 21, 2014
Example 1 – Integration with Inverse Trigonometric Functions
a)
b)
c)
Example 1 – Integration with Inverse Trigonometric Functions
u = x, a = 2
5.9 lesson filled in.notebook February 21, 2014
Completing the Square
Completing the SquareCompleting the square helps when quadratic functions are involved in the integrand.
For example, the quadratic x2 + bx + c can be written as the difference of two squares by adding and subtracting (b/2)2.
5.9 lesson filled in.notebook February 21, 2014
Example 4 – Completing the Square
Example 4 – Solution
Now, in this completed square form, let u = x – 2 and a = .
Solution:
You can write the denominator as the sum of two squares, as follows.
x2 – 4x + 7 = (x2 – 4x + 4) – 4 + 7
= (x – 2)2 + 3
5.9 lesson filled in.notebook February 21, 2014
Review of Basic Integration Rules
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.
5.9 lesson filled in.notebook February 21, 2014
Review of Basic Integration Rulescont’d
Quotient in an integral:1. Can is easily simplify (factor, cancel, etc.)?
2. Can you split it up into more than one quotient?
3. Will the answer be an ln (is the derivative of the denominator in the numerator)?
4. Is it arcsin, arctan, or arcsec?
5. Can you use usubstitution? (Remember: if an ln or an arc trig function is in an integral, it must be "u" or a part of "u.")
6. Will long division (or synthetic division) do the job?
5.9 lesson filled in.notebook February 21, 2014
Example 6 – Comparing Integration ProblemsFind as many of the following integrals as you can using the formulas and techniques you have studied so far in the text.
Example 6 – Solutiona. You can find this integral (it fits the Arcsecant Rule).
b. You can find this integral (it fits the Power Rule).
c. You cannot find this integral using the techniques you have studied so far.
5.9 lesson filled in.notebook February 21, 2014
Let’s practice some problems!Now let’s do some of the problems from section 5.9.
5.9 lesson filled in.notebook February 21, 2014
5.9 lesson filled in.notebook February 21, 2014
5.9 lesson filled in.notebook February 21, 2014
5.9 lesson filled in.notebook February 21, 2014
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