Math%104%–Calculus % 8.5%Par6al%Frac6ons% · 2015. 2. 18. · Partial Fractions! • Goal: To be...

Math 104 -‐ Yu

Math 104 – Calculus 8.5 Par6al Frac6ons

Math 104 -‐ Yu

Par6al Frac6ons

Nicolas Fraiman Math 104

Partial Fractions!

• Goal: To be able to integrate rational functions (quotients of polynomials).

• Method: Partial fraction decomposition. Write p(x)/q(x) as a sum of functions that are easy to integrate.

• Description: Write q(x) as a product of linear factors and irreducible quadratic factors. Decompose the rational function in a sum of simpler fractions.

Math 104 -‐ Yu

Easy Cases


Easy cases

•

•

•

Z1

x+ 3dx = ln |x+ 3|+ C

Z1

(x� 2)2dx =

�1

x� 2+ C

Z1

x

2 + 4dx =

1

2arctan

⇣x

2

⌘+ C

Zx

x

2+ 7

dx =

1

2

log(x

2+ 7) + C

Math 104 -‐ Yu

Review of Algebra

2/20/2014

1

Math 104 – Rimmer8.4 Partial Fraction Decomposition

Algebra Review:

( )Polynomials that can be factored over the reals are called .reducible

( )Polynomials that be factored over the reals are called .can't irreducible

2How can you tell whether is reducible?ax bx c+ +

2 24 0 is reducibleb ac ax bx c! " ⇒ + +

2 24 0 is irreducibleb ac ax bx c! < ⇒ + +

Fundamental Theorem of Algebra

Math 104 – Rimmer8.4 Partial Fraction Decomposition

2Example 1: 3 18x x+ !

( )( )2 3 18 6 3x x x x+ ! = + !

2Example 2: 4 4x x! +

( )( ) ( )22 4 4 2 2 2x x x x x! + = ! ! = !

2Example 3: 2 10x x+ !

Reducible2 4 9 72 81b ac! = + =

2 4 16 16 0b ac! = ! =

2 4 4 40 44b ac! = + =

2When 4 0 and is a perfect square, the polynomial should factor nicely because it will have rational roots.b ac! >

2When 4 0, the polynomial will have a double root.b ac! =

2When 4 0 but not a perfect square, the polynomial doesn't factor nicely because it will have irrational roots.

These rarely show up in the context of partial fractions.

b ac! >

Irreducible2Example 4: 4 13x x! + 2 4 16 52 36b ac! = ! = !

2When 4 0, the polynomial does not factor because it will have imaginary roots.b ac! <

Algebra Review:

Math 104 -‐ Yu

Fundamental Theorem of Algebra


Fundamental Theorem of Algebra!

• Every polynomial of degree n > 0, with real coefficients can be written as a product of linear and/or irreducible quadratic factors.

• is irreducible.

• Irreducible quadratic cannot be factored as a product of two linear factors.

b

2 � 4ac < 0 =) ax

2 + bx+ c




• is irreducible.


b

2 � 4ac < 0 =) ax

2 + bx+ c




• is irreducible.


b

2 � 4ac < 0 =) ax

2 + bx+ c

Math 104 -‐ Yu

Sketch of Method •  If degree of p(x) is greater than degree of q(x), do long division first. We get a sum of a quo6ent polynomial and a proper frac6on

•  Decompose the proper frac6on into a sum of par6al frac6ons:

•  Use algebra to solve for the constants (compare coefficients or evaluate at special values)

•  Integrate the quo6ent polynomial and par6al frac6ons.

x

2 + 2

x� 1= x+ 1 +

3

x� 1


Scheme of method• If degree of p(x) is greater than degree of q(x) we must

long divide first.

• Decompose q(x) into factors:LinearPowers of linearIrreducible quadratic

• Use algebra (evaluation at values) to solve for the constants.

x

2+6x�4(x�3)3 = A

x�3 + B

(x�3)2 + C

(x�3)3

6x(x+3)(2x�5) =

A

x+3 + B

2x�5

x+5(x+1)(x2+9) =

A

x+1 + Bx+C

x

2+9



long divide first.



x

2+6x�4(x�3)3 = A

x�3 + B

(x�3)2 + C

(x�3)3

6x(x+3)(2x�5) =

A

x+3 + B

2x�5

x+5(x+1)(x2+9) =

A

x+1 + Bx+C

x

2+9



long divide first.



x

2+6x�4(x�3)3 = A

x�3 + B

(x�3)2 + C

(x�3)3

6x(x+3)(2x�5) =

A

x+3 + B

2x�5

x+5(x+1)(x2+9) =

A

x+1 + Bx+C

x

2+9

Math 104 -‐ Yu

Method Descrip6on

Math 104 -‐ Yu

Examples


Examples1. Evaluate

!

2. Find

Z 2

0

x� 12

x

2 + 3x� 18dx

Z2x+ 8

x

3 � 4x2 + 4xdx

Math 104 -‐ Yu

Examples


Harder examples!

3. Find

!

!

!

4. Find

Z6x2 � 23x+ 58

(x� 2)(x2 � 4x+ 13)dx

Zx

3 � 2x2 + 18x� 29

x

2 + 16dx

Math 104 -‐ Yu

Heaviside “Cover-‐up” Method Only works

when linear factors have exponents = 1

Math%104%–Calculus % 8.5%Par6al%Frac6ons% · 2015. 2. 18. · Partial Fractions! • Goal: To be...

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