周次: 5 日期: March 25, 2022 节次: 2
In this course, we will discuss the definition of hyperbolic functions. And we shift our attention to several integration techniques. These techniques can help us integrate more functions. A typical technique called “Integration by Parts” will be discussed as well.
§ 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
§ 8.2 Integration by Parts
Integration by parts is a technique for simplifying integrals of the form
It is useful when
Product Rule in Integral Form
Integration by Parts Formula
The mindset behind the formula is that to transfer a difficult integration to an easier one.
Sometimes it is easier to remember the formula if we write it in differential form.
Why it works?
The formula for integration by parts is coming from the Product Rule for derivatives
Example 1. Using Integration by Parts
Find
Example 3. Integral of the Natural Logarithm
Find
Example 4. Repeated Use of Integration by Parts
Find
Example 5. Solving for the Unknown Integral (An Circling-Back Integration)
Find
Evaluating Definite Integrals by Parts
Combining the Fundamental Theorem of Calculus with the Product Rule, we find the following formula for definite integrals
Therefore we can find the following integration by parts formula for definite integrals.
Integration by Parts Formula for Definite Integrals
Example 6. Finding Area
Find the area of the region bounded by the curve
Example 7. Repeating Multiple Times. Ft. The Tabular Integration Method
Find
Example 9. A Reduction Formula (Iterative Formulas)
Find the integral
Example 10. Using the Reduction Formula
Find