1.26M

Category: $mathematics$ mathematics

Integrating Rational Functions by Partial Fractions

2.

Integrating Rational
Functions by Partial
Fractions
Objective: To make a difficult/impossible
integration problem easier.

3. Partial Fractions

In algebra, you learn to combine two or more
fractions into a single fraction by finding a common
denominator. For example
2
3
2( x 1) 3( x 4)
5 x 10
2
x 4 x 1
( x 4)( x 1)
x 3x 4

4. Partial Fractions

However, for the purposes of integration, the left side
of this equation is preferable to the right side since
each term is easy to integrate.
2
3
2( x 1) 3( x 4)
5 x 10
2
x 4 x 1
( x 4)( x 1)
x 3x 4
2
3
dx
x 4 x 1 dx 2 ln | x 4 | 3 ln | x 1 | C

5. Partial Fractions

We need a method to take
and make it
5 x 10
x 2 3x 4
2
3
.
x 4 x 1
This method is called Partial Fractions. This method
only works for proper rational fractions, meaning that
the degree of the numerator is less than the degree of
the denominator. This is how it works.

6. Partial Fractions

Factor the denominator completely.
5 x 10
5 x 10
2
x 3x 4 ( x 4)( x 1)

7. Partial Fractions

Factor the denominator completely.
5 x 10
5 x 10
2
x 3x 4 ( x 4)( x 1)
Assign a variable as the numerator to each term of
the denominator and set it equal to the original.
5 x 10
A
B
( x 4)( x 1) x 4 x 1

8. Partial Fractions

Multiply by the common denominator.
5 x 10
A
B
( x 4)( x 1) x 4 x 1
5 x 10 A( x 1) B( x 4)

9. Partial Fractions

Multiply by the common denominator.
5 x 10
A
B
( x 4)( x 1) x 4 x 1
51
A ) B ( x 4)
5 x 10 A( x10
2 A
Solve for A and B.
To solve for A, let x = 4, which gives us
To solve for B, let x = -1, which gives us 15 5B
3 B
2
3
dx
x 4 x 1 dx 2 ln | x 4 | 3 ln | x 1 | C

10. Example 1

Evaluate
dx
x2 x 2

11. Example 1

Evaluate
dx
x2 x 2
dx
dx
x 2 x 2 ( x 2)( x 1)

12. Example 1

Evaluate
dx
x2 x 2
dx
dx
x 2 x 2 ( x 2)( x 1)
1
A
B
( x 2)( x 1) x 2 x 1

13. Example 1

Evaluate
dx
x2 x 2
dx
dx
x 2 x 2 ( x 2)( x 1)
1 A( x 1) B( x 2)
1
A
B
( x 2)( x 1) x 2 x 1

14. Example 1

Evaluate
dx
x2 x 2
dx
dx
x 2 x 2 ( x 2)( x 1)
1 A( x 1) B( x 2)
x 1
1 3B
1/ 3 B
x 2
1 3 A
1/ 3 A
1
A
B
( x 2)( x 1) x 2 x 1

15. Example 1

Evaluate
dx
x2 x 2
1 1
1
1
1
1
ln | x 1 | ln | x 2 |
3 x 1 3 x 2 3
3
1 x 1
ln
C
3 x 2

16. Linear Factors

Linear Factor Rule.
For each factor of the form (ax b) , the partial
m
fractions decomposition contains the following sum of
m partial fractions
Am
A1
A2
...
2
ax b (ax b)
(ax b) m
where A1, A2, …Am are constants to be determined. In
the case where m = 1, only the first term appears.

17. Example 2

Evaluate
2x 4
x3 2 x 2 dx

18. Example 2

Evaluate
2x 4
x3 2 x 2 dx
2x 4
x 2 ( x 2)

19. Example 2

Evaluate
2x 4
x3 2 x 2 dx
2x 4
x 2 ( x 2)
2x 4
A B
C
2
x x( x 2) x x
x 2

20. Example 2

Evaluate
2x 4
x3 2 x 2 dx
2x 4
x 2 ( x 2)
2x 4
A B
C
2
x x( x 2) x x
x 2
2 x 4 Ax( x 2) B( x 2) Cx 2

21. Example 2

Evaluate
2x 4
x3 2 x 2 dx
2x 4
x 2 ( x 2)
2x 4
A B
C
2
x x( x 2) x x
x 2
2 x 4 Ax( x 2) B( x 2) Cx 2
x 0
4 2 B
2 B
x 2
8 4c
2 C

22. Example 2

Evaluate
2x 4
x3 2 x 2 dx
x 2
x 0
2 x 4 Ax( x 2) B( x 2) Cx 2 4 2 B
8 4c
2 B
2 C
Since there is no way to isolate A, we need to solve
with a different method. Let x = 1, substitute our
values for B and C and solve for A.

23. Example 2

24. Example 2

Evaluate
2x 4
x3 2 x 2 dx
dx
dx
dx
2 2 2 2
x
x
x 2
2
2 ln | x | 2 ln | x 2 | C
x
2
x 2
2 ln |
| C
x
x
x 0
4 2 B
2 B
x 2
8 4c
2 C
6 A 4
2 A

25. Example 2-WRONG

Evaluate
2x 4
x3 2 x 2 dx
2x 4
x 2 ( x 2)
2x 4
A B
C
x x( x 2) x x x 2
2 x 4 Ax( x 2) Bx( x 2) Cx 2

26. Example 2-WRONG

Evaluate
2x 4
x3 2 x 2 dx
2x 4
x 2 ( x 2)
2x 4
A B
C
x x( x 2) x x x 2
2 x 4 Ax( x 2) Bx( x 2) Cx 2
What next? This doesn’t work!
x 2
8 4c
2 C

27. Example 2-WRONG

Evaluate
2x 4
x3 2 x 2 dx
2x 4
x 2 ( x 2)
2x 4
A B
C
x x( x 2) x x x 2
The denominator on the right is
The denominator on the left is
They are not the same!!
x( x 2)
x 2 ( x 2)

28. Quadratic Factors

Quadratic Factor Rule
2
m
(
ax
bx
c
)
For each factor of the form
, the partial
fraction decomposition contains the following sum of
m partial fractions:
Am x Bm
A1 x B1
A2 x B2
...
2
2
2
ax bx c (ax bx c)
(ax 2 bx c) m
where A1, A2,…Am, B1, B2,…Bm are constants to be
determined. In the case where m = 1, only the first
term appears.

29. Example 3

Evaluate
x2 x 2
3x3 x 2 3x 1dx

30. Example 3

Evaluate
x2 x 2
3x3 x 2 3x 1dx
Factor by grouping
x 2 (3 x 1) 1(3 x 1) (3 x 1)( x 2 1)
x2 x 2
A
Bx C
2
2
(3 x 1)( x 1) 3 x 1 x 1

31. Example 3

Evaluate
x2 x 2
3x3 x 2 3x 1dx
Factor by grouping
x 2 (3 x 1) 1(3 x 1) (3 x 1)( x 2 1)
x2 x 2
A
Bx C
2
2
(3 x 1)( x 1) 3 x 1 x 1
x 2 x 2 A( x 2 1) ( Bx C )(3 x 1)

32. Example 3

Evaluate
x2 x 2
3x3 x 2 3x 1dx
Multiply the right side of the equation and group the
terms based on powers of x.
x 2 x 2 A( x 2 1) ( Bx C )(3 x 1)
x 2 x 2 Ax 2 A 3Bx 2 Bx 3Cx C
x 2 x 2 ( A 3B) x 2 (3C B) x ( A C )

33. Example 3

Evaluate
x2 x 2
3x3 x 2 3x 1dx
Set the coefficients from the right side of the equation
equal to the ones on the left side.
x 2 x 2 ( A 3B) x 2 (3C B) x ( A C )
1 A 3B
1 3C B
2 A C

34. Example 3

Evaluate
x2 x 2
3x3 x 2 3x 1dx
Take Eq 1 – Eq 3
1 A 3B
1 3C B
2 A C
1 A 3B
( 2 A C )
3 C 3B

35. Example 3

Evaluate
x2 x 2
3x3 x 2 3x 1dx
Take Eq 1 – Eq 3
Take new Eq + 3Eq 2
1 A 3B
1 3C B
2 A C
1 A 3B
( 2 A C )
3 C 3B
3 C 3B
3 9C 3B
6 10C
3/ 5 C

36. Example 3

Evaluate
x2 x 2
3x3 x 2 3x 1dx
Take Eq 1 – Eq 3
Take new Eq + 3Eq 2
1 A 3B
1 3C B
2 A C
1 A 3B
( 2 A C )
3 C 3B
3 9C 3B
3 C 3B
6 10C
3/ 5 C
7/5 A
4/5 B

37. Example 3

Evaluate
x2 x 2
3x3 x 2 3x 1dx
x2 x 2
7/5
(4 / 5) x 3 / 5
dx
dx
dx
2
2
(3 x 1)( x 1)
3x 1
x 1
x2 x 2
7/5
(4 / 5) x
3/ 5
dx
dx
dx 2
dx
2
2
(3x 1)( x 1)
3x 1
x 1
x 1

38. Example 3

Evaluate
x2 x 2
3x3 x 2 3x 1dx
x2 x 2
7/5
(4 / 5) x 3 / 5
dx
dx
dx
2
2
(3 x 1)( x 1)
3x 1
x 1
x2 x 2
7/5
(4 / 5) x
3/ 5
dx
dx
dx 2
dx
2
2
(3x 1)( x 1)
3x 1
x 1
x 1
7
2
3
2
ln | 3x 1 | ln( x 1) tan 1 x C
15
5
5

39. Example 4

Evaluate
3x 4 4 x 3 16 x 2 20 x 9
dx
2
2
( x 2)( x 3)

40. Example 4

Evaluate
3x 4 4 x 3 16 x 2 20 x 9
dx
2
2
( x 2)( x 3)
3 x 4 4 x 3 16 x 2 20 x 9
A
Bx C Dx E
2
2
2
2
( x 2)( x 3)
x 2 x 3 ( x 3) 2

49. Homework

From 17-50 :
Only Odd numbers

English Русский Rules