Section 1



Section 6.4: Partial Fractions

Practice HW from Larson Textbook (not to hand in)

p. 399 # 1-19 odd

Partial Fractions

Decomposes a rational function into simpler rational functions that are easier to integrate. Essentially undoes the process of finding a common denominator of fractions.

Partial Fractions Process

1. Check to make sure the degree of the numerator is less than the degree of the

denominator. If not, need to divide by long division.

2. Factor the denominator into linear or quadratic factors of the form

Linear: [pic] Quadratic: [pic]

3. For linear functions:

[pic] where

[pic] are real numbers.

4. For Quadratic Factors:

[pic]

where [pic] and [pic].

Integrating Functions With Linear Factors Using Partial Fractions

1. Substitute the roots of the distinct linear factors of the denominator into the basic equation (the equation obtained after eliminating the fractions on both sides of the equation) and find the resulting constants.

2. For repeated linear factors, use the coefficients found in step 1 and substitute other

convenient values of x to find the other coefficients.

3. Integrate each term.

Example 1: Integrate [pic]

Solution:

█

Example 2: Integrate [pic]

Solution: Note that the denominator

[pic]

and hence the linear factors has roots of [pic] and [pic]. We perform the partial fraction expansion as follows:

[pic]

To eliminate the fractions and obtain the basic equation, we multiply both sides by [pic] and obtain

[pic]

[pic]

[pic] (Basic Equation)

To find A and C, we substitute the roots of the linear factors, x = 0 and x = -3, of the denominator into the basic equation. This gives

[pic]

(continued on next page)

[pic]

The basic equation is defined for all values of x. To find B, since we only have linear factors, we can choose a random value of x to substitute into the basic equation. An easy value to choose is [pic]. This gives

[pic]

Substituting A = -1, B = 1, and C = 4 into

[pic]

(continued on next page)

gives

[pic]

or

[pic].

Integrating, we obtain

[pic]

Let u = x+ 3 u = x+ 3

du = dx du = dx

[pic]

█

Integrating terms using Partial Fractions with Irreducible Quadratic Terms [pic] (quadratic terms that cannot be factored) in the Denominator.

1. Expand the basic equation and combine the like terms of x.

2. Equate the coefficients of like powers and solve the resulting system of equations.

3. Integrate.

Example 3: Integrate [pic]

Solution:

█

Example 4: Integrate [pic]

Solution: Note that the denominator

[pic]

and hence the roots of the linear factors are x = -2 and x = 2. We perform the partial fraction expansion as follows:

[pic]

To clear the equation of fractions and obtain the basic equation, we multiply both sides of this equation by [pic]. This gives

[pic]

[pic]

[pic] (Basic Equation)

To find A and B, we substitute the roots of the linear factors, x = 2 and x = -2, of the denominator into the basic equation. This gives

[pic]

(Continued on Next Page)

[pic]

To find C and D, we must substitute the values of A and B we found into the basic equation, multiply out, and equate like coefficients.

[pic]

We write the equation in the following form and equate like coefficients.

(Continued on Next Page)

[pic]

Equating like coefficients gives

[pic], [pic], [pic], [pic]

Substituting [pic], [pic], [pic], and [pic] into

[pic]

gives

[pic]

(Continued on Next Page)

Integrating, we obtain

[pic]

Let u = x+ 2 u = x- 2 [pic]

du = dx du = dx [pic]

[pic]

[pic]

█

Example 5: Find the partial fraction expansion of

[pic]

Solution:

█

Integration Technique Chart

[pic]

-----------------------

Basic

Formula

u-du栨株栬棦棨椎椐椒椔検椞椦椨楎楐楒楔楰楴楶榖榘榬ퟦ꪿蚕瞿虣垿䑉㨿ᘉ橨ٓ㔀脈ᘉ���脱㔀脈ᘉﱨ簽㔀脈ᔚ콨ᘀἋ㔀脈࠶䎁ᑊ愀ᑊᔗ偨訛ᘀ偨訛㔀脈䩃䩡̧j£ᔀ콨ᘀ器塘㔀脈࠶䎁ᑊ䔀ш嗿Ĉ䩡̝ⱪ軡੍Ĉ栖塖X䩃ࡕ嘁Ĉ䩡̝jᘀ偨訛㔀脈࠶䎁ᑊ唀Ĉ䩡̨jᘀၨ轾㔀脈࠶䎁ᑊ唀Ĉ䩡䡭Ѐ䡮Ѐࡵ[pic]̨jᘀ콨㔀脈࠶䎁ᑊ唀Ĉ䩡䡭Ѐ䡮Ѐࡵ[pic]ᘔ偨訛㔀脈࠶䎁ᑊ愀ᑊ̙ꥪ ᔀ偨訛ᘀ偨訛䔀嗿Ĉ̝멪ᘿ੍Ĉ栖᭐Š䩃ࡕ嘁Ĉ䩡ᘆ偨訛̏j Substitution: Look for term to set u = to whose derivative similar to another term

Terms Multiplied No Denominators

Integration by Parts

Sine and

Cosine Integral Techniques

Trig Substitution

Partial

Fractions

Sine anc Cosine Powers

Radicals

Fractional

Expressions

Function to

Integrate

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