Lecture17 1 Integration by partial fractionspioneer.netserv.chula.ac.th/~ksujin/slide17(ISE).pdf ·...
Transcript of Lecture17 1 Integration by partial fractionspioneer.netserv.chula.ac.th/~ksujin/slide17(ISE).pdf ·...
![Page 1: Lecture17 1 Integration by partial fractionspioneer.netserv.chula.ac.th/~ksujin/slide17(ISE).pdf · Lecture17| 1 Integration by partial fractions To integrate the fraction one cannot](https://reader030.fdocuments.in/reader030/viewer/2022040618/5f291cc2c77b6b36e143c4d6/html5/thumbnails/1.jpg)
Le c t u r e 1 7 | 1
Integration by partial fractions To integrate the fraction
one cannot use all the previous methods (substitution, by parts).
However, one observe that the fraction can be reduced into
i.e. it is the sum of other fractions. Then
![Page 2: Lecture17 1 Integration by partial fractionspioneer.netserv.chula.ac.th/~ksujin/slide17(ISE).pdf · Lecture17| 1 Integration by partial fractions To integrate the fraction one cannot](https://reader030.fdocuments.in/reader030/viewer/2022040618/5f291cc2c77b6b36e143c4d6/html5/thumbnails/2.jpg)
Le c t u r e 1 7 | 2
In this section we study the technique to integrate fractions
We will see that the fraction
can be reduced into sum (or difference) of other fractions called partial fractions.
This is called the partial fractions expansion of .
Then the integral
is the sum
of the integrals of partial fractions.
![Page 3: Lecture17 1 Integration by partial fractionspioneer.netserv.chula.ac.th/~ksujin/slide17(ISE).pdf · Lecture17| 1 Integration by partial fractions To integrate the fraction one cannot](https://reader030.fdocuments.in/reader030/viewer/2022040618/5f291cc2c77b6b36e143c4d6/html5/thumbnails/3.jpg)
Le c t u r e 1 7 | 3
EX
![Page 4: Lecture17 1 Integration by partial fractionspioneer.netserv.chula.ac.th/~ksujin/slide17(ISE).pdf · Lecture17| 1 Integration by partial fractions To integrate the fraction one cannot](https://reader030.fdocuments.in/reader030/viewer/2022040618/5f291cc2c77b6b36e143c4d6/html5/thumbnails/4.jpg)
Le c t u r e 1 7 | 4
If we factorize
we call
In the case (or ), the term is called simple.
If (or ), the term is said to be repeated with order (resp., ).
![Page 5: Lecture17 1 Integration by partial fractionspioneer.netserv.chula.ac.th/~ksujin/slide17(ISE).pdf · Lecture17| 1 Integration by partial fractions To integrate the fraction one cannot](https://reader030.fdocuments.in/reader030/viewer/2022040618/5f291cc2c77b6b36e143c4d6/html5/thumbnails/5.jpg)
Le c t u r e 1 7 | 5
EX The polynomial
has a simple linear term , a repeated linear term , order a repeated quad term ,
order 2
![Page 6: Lecture17 1 Integration by partial fractionspioneer.netserv.chula.ac.th/~ksujin/slide17(ISE).pdf · Lecture17| 1 Integration by partial fractions To integrate the fraction one cannot](https://reader030.fdocuments.in/reader030/viewer/2022040618/5f291cc2c77b6b36e143c4d6/html5/thumbnails/6.jpg)
Le c t u r e 1 7 | 6
Partial Fraction (with Long division)
If , using long division we can express
where . So
can be easily integrated.
If then . So long division is not required.
![Page 7: Lecture17 1 Integration by partial fractionspioneer.netserv.chula.ac.th/~ksujin/slide17(ISE).pdf · Lecture17| 1 Integration by partial fractions To integrate the fraction one cannot](https://reader030.fdocuments.in/reader030/viewer/2022040618/5f291cc2c77b6b36e143c4d6/html5/thumbnails/7.jpg)
Le c t u r e 1 7 | 7
EX Use long division to reduce
Then integrate
![Page 8: Lecture17 1 Integration by partial fractionspioneer.netserv.chula.ac.th/~ksujin/slide17(ISE).pdf · Lecture17| 1 Integration by partial fractions To integrate the fraction one cannot](https://reader030.fdocuments.in/reader030/viewer/2022040618/5f291cc2c77b6b36e143c4d6/html5/thumbnails/8.jpg)
Le c t u r e 1 7 | 8
Now we have to study
![Page 9: Lecture17 1 Integration by partial fractionspioneer.netserv.chula.ac.th/~ksujin/slide17(ISE).pdf · Lecture17| 1 Integration by partial fractions To integrate the fraction one cannot](https://reader030.fdocuments.in/reader030/viewer/2022040618/5f291cc2c77b6b36e143c4d6/html5/thumbnails/9.jpg)
Le c t u r e 1 7 | 9
Partial Fraction: (simple linear terms) If
where are distinct real numbers, then there are such that
can be computed from
Formula
![Page 10: Lecture17 1 Integration by partial fractionspioneer.netserv.chula.ac.th/~ksujin/slide17(ISE).pdf · Lecture17| 1 Integration by partial fractions To integrate the fraction one cannot](https://reader030.fdocuments.in/reader030/viewer/2022040618/5f291cc2c77b6b36e143c4d6/html5/thumbnails/10.jpg)
Le c t u r e 1 7 | 10
EX Find the partial fraction expansion of
Then integrate
![Page 11: Lecture17 1 Integration by partial fractionspioneer.netserv.chula.ac.th/~ksujin/slide17(ISE).pdf · Lecture17| 1 Integration by partial fractions To integrate the fraction one cannot](https://reader030.fdocuments.in/reader030/viewer/2022040618/5f291cc2c77b6b36e143c4d6/html5/thumbnails/11.jpg)
Le c t u r e 1 7 | 11
EX Find the partial fraction expansion of
Then evaluate
![Page 12: Lecture17 1 Integration by partial fractionspioneer.netserv.chula.ac.th/~ksujin/slide17(ISE).pdf · Lecture17| 1 Integration by partial fractions To integrate the fraction one cannot](https://reader030.fdocuments.in/reader030/viewer/2022040618/5f291cc2c77b6b36e143c4d6/html5/thumbnails/12.jpg)
Le c t u r e 1 7 | 12
Generally, if the factorization of has a simple linear term (together with others, which could be repeated, or quadratics), this term gives rise to
in the partial fraction expansion of
is computed by the same formula:
![Page 13: Lecture17 1 Integration by partial fractionspioneer.netserv.chula.ac.th/~ksujin/slide17(ISE).pdf · Lecture17| 1 Integration by partial fractions To integrate the fraction one cannot](https://reader030.fdocuments.in/reader030/viewer/2022040618/5f291cc2c77b6b36e143c4d6/html5/thumbnails/13.jpg)
Le c t u r e 1 7 | 13
Partial Fraction: (repeated linear term)
If the factorization of has
then this term gives rise to
in the partial fraction expansion of
.
can be computed from
Formula
![Page 14: Lecture17 1 Integration by partial fractionspioneer.netserv.chula.ac.th/~ksujin/slide17(ISE).pdf · Lecture17| 1 Integration by partial fractions To integrate the fraction one cannot](https://reader030.fdocuments.in/reader030/viewer/2022040618/5f291cc2c77b6b36e143c4d6/html5/thumbnails/14.jpg)
Le c t u r e 1 7 | 14
EX Find the partial fraction expansion of
Then evaluate the integral
![Page 15: Lecture17 1 Integration by partial fractionspioneer.netserv.chula.ac.th/~ksujin/slide17(ISE).pdf · Lecture17| 1 Integration by partial fractions To integrate the fraction one cannot](https://reader030.fdocuments.in/reader030/viewer/2022040618/5f291cc2c77b6b36e143c4d6/html5/thumbnails/15.jpg)
Le c t u r e 1 7 | 15
EX Evaluate
![Page 16: Lecture17 1 Integration by partial fractionspioneer.netserv.chula.ac.th/~ksujin/slide17(ISE).pdf · Lecture17| 1 Integration by partial fractions To integrate the fraction one cannot](https://reader030.fdocuments.in/reader030/viewer/2022040618/5f291cc2c77b6b36e143c4d6/html5/thumbnails/16.jpg)
Le c t u r e 1 7 | 16
Partial Fraction: (simple quad terms)
If the factorization of has a simple quad term
then this term gives rise to
in the partial fraction expansion of
.
Formula
Repeated quad terms need the trig. subst., will be discussed later.
![Page 17: Lecture17 1 Integration by partial fractionspioneer.netserv.chula.ac.th/~ksujin/slide17(ISE).pdf · Lecture17| 1 Integration by partial fractions To integrate the fraction one cannot](https://reader030.fdocuments.in/reader030/viewer/2022040618/5f291cc2c77b6b36e143c4d6/html5/thumbnails/17.jpg)
Le c t u r e 1 7 | 17
EX Find the partial fraction expansion of
Then evaluate the integral
![Page 18: Lecture17 1 Integration by partial fractionspioneer.netserv.chula.ac.th/~ksujin/slide17(ISE).pdf · Lecture17| 1 Integration by partial fractions To integrate the fraction one cannot](https://reader030.fdocuments.in/reader030/viewer/2022040618/5f291cc2c77b6b36e143c4d6/html5/thumbnails/18.jpg)
Le c t u r e 1 7 | 18
EX (Change of variable first) Evaluate
![Page 19: Lecture17 1 Integration by partial fractionspioneer.netserv.chula.ac.th/~ksujin/slide17(ISE).pdf · Lecture17| 1 Integration by partial fractions To integrate the fraction one cannot](https://reader030.fdocuments.in/reader030/viewer/2022040618/5f291cc2c77b6b36e143c4d6/html5/thumbnails/19.jpg)
Le c t u r e 1 7 | 19
EX (Change of variable first) Evaluate