Integration, Partial Fraction Decomposition - Notes | MATH 250A, Study notes of Differential Equations

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MATH 250aFall Semester 2007Section 2 (J. M. Cushing) Thursday, September 27

http://math.arizona.edu/~cushing/250a.html

Chapter 7: Integration^ Sections 1 - 4

¾^ Calculation of anti-derivatives (integrals)^ 1. Learn some fundamental properties of integrals 2. Use known differentiation formulas “backwards” 3. Use differentiation rules “backwards”chain rule

substitution product rule

integration by parts

4.^ Use algebraic manipulations

e.g., partial fraction decomposition

9 9 9 5.^ Use tables of integrals

4. Use algebraic manipulations: Partial Fraction Decomposition

Useful for integrating rational functions. First, we look at some simpler rational functions:

4. Use algebraic manipulations: Partial Fraction Decomposition

Useful for integrating rational functions. First, we look at some simpler rational functions :^1

1

ln |^ | dx^

dw^

w^ c

x^ a^

=^ =^ w

∫^ −

∫

4. Use algebraic manipulations: Partial Fraction Decomposition

Useful for integrating rational functions. First, we look at some simpler rational functions :^1

1

ln |^ | dx^

dw^

w^ c

x^ a^

=^ =^ w

∫^ −

∫ dw , 1 w^ x^

a^

dw^ dx dx =^ −^

=^ ⇒^

=

4. Use algebraic manipulations: Partial Fraction Decomposition

Useful for integrating rational functions. First, we look at some simpler rational functions :^1

1

ln |^ | dx^

dw^

w^ c

x^ a^

=^ =^ w

∫^ −

∫ dw , 1 w^ x^

a^

dw^ dx dx =^ −^

=^ ⇒^

=

4. Use algebraic manipulations: Partial Fraction Decomposition

Useful for integrating rational functions. First, we look at some simpler rational functions : 1

ln |^

dx^

x^ a^

c

x^ a^

=^

−^ +

∫−

4. Use algebraic manipulations: Partial Fraction Decomposition

Useful for integrating rational functions. First, we look at some simpler rational functions :

1 n ( ) dx (^) x a− ∫

,^ du^1 u^ x^

a^

du^ dx dx =^ +

=^ ⇒

=

for an integer

n^ > 1

ln |^

dx^

x^ a^

c

x^ a^

=^

−^ +

∫−

4. Use algebraic manipulations: Partial Fraction Decomposition

Useful for integrating rational functions. First, we look at some simpler rational functions :

(^1) ( )

1 n dx^ n

dww x^ a^

∫^ −

∫

ln |^

dx^

x^ a^

c

x^ a^

=^

−^ +

∫−

,^ dw^1 w^ x^

a^

dw^ dx dx =^ −^

=^ ⇒^

=

4. Use algebraic manipulations: Partial Fraction Decomposition

Useful for integrating rational functions. First, we look at some simpler rational functions :

(^ )

1

1 n 1 n dx^ n

dw^

w^ c w^

n

x^ a

−

=^

=^

∫^ −

∫

ln |^

dx^

x^ a^

c

x^ a^

=^

−^ +

∫−

,^ dw^1 w^ x^

a^

dw^ dx dx =^ −^

=^ ⇒^

=

4. Use algebraic manipulations: Partial Fraction Decomposition

Useful for integrating rational functions. First, we look at some simpler rational functions :

(^ )^

(^ )^

1

1

n dx^ n

c n x^ a^

−x a =^

−^

∫

ln |^

dx^

x^ a^

c

x^ a^

=^

−^ +

∫−

4. Use algebraic manipulations: Partial Fraction Decomposition

Useful for integrating rational functions. First, we look at some simpler rational functions : 1 dx (^2 2) x a+ ∫

(^ )^

(^ )^

1

1

n dx^ n

c n x^ a^

−x a =^

−^

∫

ln |^

dx^

x^ a^

c

x^ a^

=^

−^ +

∫−

4. Use algebraic manipulations: Partial Fraction Decomposition

Useful for integrating rational functions. First, we look at some simpler rational functions :

2 2

2

2 2

2 2 1

x^

x a^

a

dx^

dx^

dx

x^ a^

a

=^ a

⎛^ ⎞

+^

⎛^ ⎞

⎜^

⎝^ ⎠

⎝^ ⎠

∫^

∫^

∫ (^ )^

(^ )^

1

1

n dx^ n

c n x^ a^

−x a =^

−^

∫

ln |^

dx^

x^ a^

c

x^ a^

=^

−^ +

∫−

Recall:^

arctandx x 1

c x^

=^

∫+

4. Use algebraic manipulations: Partial Fraction Decomposition

Useful for integrating rational functions. First, we look at some simpler rational functions :

2 2 2

2 1

(^1) x^1 a dx^

dx

x^ a^

=a +^

⎛^ ⎞⎟⎜^ +⎟⎜^ ⎟⎜⎝^ ⎠

∫^

∫

(^ )^

(^ )^

1

1

n dx^ n

c n x^ a^

−x a =^

−^

∫

ln |^

dx^

x^ a^

c

x^ a^

=^

−^ +

∫−

Recall:^

arctandx x 1

c x^

=^

∫+