Math 252 Calculus 2 Chapter 8 Section 2 Completed
Integration by Parts
Integration by Parts is a technique of integration that is useful when the integrand involves a product of an algebraic with a transcendental expression, such as …
[pic] [pic] [pic]
See Exercise 1. See Exercise 2. See Exercise 5.
Integration by Parts is based on the Product Rule for taking derivatives.
Product Rule: [pic] = where u and v are functions of x
If [pic] and [pic] are continuous, then you can integrate both sides of the Product Rule equation.
[pic] = [pic] + [pic]
This produces: = where dv = and du =
Subtracting[pic] from both sides of the previous equation, we obtain the following:
*** ***
This last equation that we just created is the Integration by Parts formula. [pic] is the “original integral,” the one that we are trying to integrate. Notice that it is written in terms of another integral,[pic]. Choosing u and dv wisely will make it easier to determine the integral[pic] than it would be to determine the original integral,[pic]. Let’s look at some guidelines for choosing u and dv well. (“You have chosen…wisely.” This quote is from what movie?)
Guidelines for Choosing u and dv
1. Try letting dv be the most “complicated” portion of the integrand that fits an elementary integration rule. Then u will be the remaining factor(s) of the integrand.
2. Try letting u be the portion of the integrand whose derivative is a function that is “simpler” than u. The dv will be the remaining factor(s) of the integrand.
Exercise 1: Determine[pic] .
Let u = and let dv =
Then du = and v = [pic] =
Thus, [pic] =
Exercise 2: Determine[pic] .
Fruitless First Attempt at Exercise 2
Hmmm. If we let [pic], then we are forced to let [pic].
Okay, du will be easy to compute. But then v = [pic] = [pic] = ???
I hope and pray…that you did not say…that[pic] equals [pic]… for you’d be astray!
|[pic] is the derivative of ln(x). |[pic] is NOT an anti-derivative of ln(x). |
Fruitful Second Attempt at Exercise 2
Let u = and let dv =
Then du = and v = [pic] =
Thus, [pic] =
An Integrand with a Single Factor
A surprising use of Integration by Parts involves some problems in which the integrand consists of a single factor. Bonusly (a new adverb…and a near-antonym of “bogusly”), we’ll also solve the problem raised in the Fruitless First Attempt at Exercise 2.
Exercise 3: Determine[pic]. In these sorts of cases, let dv = .
Let u = and let dv =
Then du = and v = [pic] =
Thus, [pic] =
Check this answer to see that you are right by derivating our anti-derivative answer.
[pic] = [pic] - [pic] =
Repeated Use of Integration by Parts
Some integrals require repeated use of the Integration by Parts formula, such as…
Exercise 4: Determine[pic] . (Compare with Exercise 1.)
The derivative of becomes “simpler,” whereas the derivative of does not.
Therefore, let u = and let dv =
Then du = and v =
Thus, [pic] = [pic] - [pic] .
We now have to integrate[pic] by parts, with a new u and a new dv.
Let new u = and let new dv =
Then new du = and new v =
Thus, [pic] =
Finally,[pic] = [pic] -
=
A Clever Trick
Exercise 5: Determine[pic] . Let u = [pic] du =
Let dv = [pic] v =
Thus,[pic] =
Let new u = [pic] new du =
Let new dv = [pic] new v =
Then, [pic] =
[pic] =
Tabular Method
For problems involving repeated applications of Integration by Parts, a tabular method can help to organize the work. This method works well for integrands with a factor that is a power of x, such as…
[pic] [pic] [pic]
Exercise 6: Determine[pic] . Use the tabular method.
Let u = and let dv =
| | | | |dv = [pic] and its |
|Alternating Signs | |u and its derivatives | |anti-derivatives |
| | | | | |
|+ | |[pic] | |cos(2x) |
| | | | | |
|- | | | | |
| | | | | |
|+ | | | | |
| | | | | |
|- | | | | |
| | | | | |
| | | | | |
| | | | | |
| | | | | |
| | | | | |
| | | | | |
Thus, [pic] =
................
................
In order to avoid copyright disputes, this page is only a partial summary.
To fulfill the demand for quickly locating and searching documents.
It is intelligent file search solution for home and business.
Related searches
- chapter 8 photosynthesis 8 2
- chapter 8.2 photosynthesis overview answers
- chapter 8 section 2 photosynthesis
- chapter 8 section 2 photosynthesis answers
- chapter 8 2 photosynthesis overview answers
- chapter 4 section 2 economics
- go math chapter 8 review
- chapter 8 lesson 2 homework
- chapter 8 lesson 2 homework elements and chemical bonds
- ch chapter 8 section 1 starting a business quiz
- physical science chapter 19 nonmetal section 2 study guide
- chapter 8 section 8 2