周次: 5 日期: March 23, 2022 节次: 1
In this lecture, we define the inverse trigonometric functions, and develop some basic properties. The properties we are focusing on are the differentiation and integration properties. A brief introduction on hyperbolic functions, in particular the hyperbolic cosine and hyperbolic sine functions are given.
As the second half, we discuss some integration techniques. Those techniques could help us to find more anti-derivatives and as a result, evaluate more definite integrals.
§ 7.7 Inverse Trigonometric Functions
The six basic trigonometric functions are not one-to-one. However, we can restrict their domain to intervals on which they are one-to-one. The sine function increases from
The cosine function is one-to-one on
The tangent function is one-to-one on
The cotangent function is one-to-one on
The secant function is one-to-one on
The cosecant function is one-to-one on
Those restricted trigonometric functions have inverses, denoted by (arc). So we have
The Arcsine and Arccosine Functions
Definition. Arcsine and Arccosine Functions
is the angle in whose sine is . is the angle in whose cosine is .
Sine (blue) and arcsine (yellow) functions
The graph of
Cosine (blue) and arccosine (yellow) functions
The graph of
Example 1. Common values of arcsine function
; ; ;
Example 2. Common values of arccosine function
; ; ; .
Identities Involving Arcsine and Arccosine
Basic Identities
Since
Since
Inverses of , , ,and
Definition. Arctangent and Arccotangent Function
is the angle in for which is the angle in for which
The graph of arctan(x)
The graph of
The graph of arccot(x)
Example 3. Common Values of Arctangent Function
;
Example 4. Find , , , if
Example 5. Find
The Derivative of Arcsine Function
Find
The Chain Rule shows
Example 7. Applying the Derivative Formula
The Derivative of Arctangent Function
Find
Derivatives of the Other Two
Inverse Function-Inverse Cofunction Identities
Thus
Example 10. A Tangent Line to the Arccotangent Curve
Find an equation for the line tangent to the graph of
Derivatives Table
Integration Formulas
The Integration Formulas
For integrations like
Similarly,
Example 11. Using the Integral Formulas
Example 13. Completing the Square
- Evaluate
- Evaluate
Example 15. Using Substitution
Evaluate
§ 7.8 A Brief Introduction on Hyperbolic Functions
Let’s recall the odd and even functions. We have stated that for every function
$$ f(x) = \underbrace{\frac{f(x) + f(-x)}{2}}{\text{even part}} + \underbrace{\frac{f(x)-f(-x)}{2}}{\text{odd part}} $$
No exception for exponential function
Graphs of Hyperbolic Functions
Identities for Hyperbolic Functions
Other Hyperbolic Functions
Analogous to trigonometric functions, the
The Derivative and Anti-derivative of Hyperbolic Functions
Therefore the anti-derivatives read
Chapter 8. Techniques of Integration
In this chapter we study a number of important techniques for finding indefinite integrals of more complicated functions
§ 8.1 Basic Integration Formulas
Try to match integrals that confront us against one of the standard types. This usually involves a certain amount of algebraic manipulation as well as use of the substitution rules.
Recall that the substitution rule is
where
**Note
** Students should have a good mastery of formulas 12, 13, 18, 19. For 20 and 22, we provide an alternative (equivalent) anti-derivative
For 20, we have an alternative way to show the anti-derivative, by applying the integration table.
Example 1. Making a Simplifying Substitution
Evaluate
Example 2. Completing the Square
Evaluate
Example 3. Expanding a Power and Using a Trigonometric Identity
Evaluate
Example 4. Eliminating a Square Root
Evaluate
Example 5. Reducing an Improper Fraction
Evaluate
Example 6. Separating a Fraction
Evaluate
Example 7. Integral of — Multiplying by a Form of
Evaluate
Similarly, one can find