1. 7.4 – Integration of Rational Functions by Partial Fractions
Math 181
7.4 – Integration of Rational Functions by Partial Fractions

2. Let’s look at a technique to break rational functions up into the sum of simpler rational functions. For example: 5𝑥−3 𝑥 2 −2𝑥−3 = 5𝑥−3 𝑥+1 𝑥−3 = 2 𝑥 𝑥−3 This technique is called the method of partial fractions.

3. Ex 1. Expand the quotient by partial fractions. 𝑥 2 +4𝑥+1 𝑥−1 𝑥+1 𝑥+3

4. Ex 2. Evaluate the following integral. 𝑥 2 +4𝑥+1 𝑥−1 𝑥+1 𝑥+3 𝑑𝑥

5. In general, we need to factor the denominator first
In general, we need to factor the denominator first. What happens if we don’t get linear factors?
If we get a repeated linear factor 𝒙−𝒓 𝒎 , then we’ll have corresponding partial fractions: 𝑨 𝟏 𝒙−𝒓 + 𝑨 𝟐 𝒙−𝒓 𝟐 +…+ 𝑨 𝒎 𝒙−𝒓 𝒎
If we get an irreducible quadratic factor 𝒙 𝟐 +𝒑𝒙+𝒒, then we’ll have a corresponding partial fraction: 𝑨𝒙+𝑩 𝒙 𝟐 +𝒑𝒙+𝒒
If we get a repeated irreducible quadratic factor 𝒙 𝟐 +𝒑𝒙+𝒒 𝒏 , then we’ll have corresponding partial fractions: 𝑩 𝟏 𝒙+ 𝑪 𝟏 𝒙 𝟐 +𝒑𝒙+𝒒 + 𝑩 𝟐 𝒙+ 𝑪 𝟐 𝒙 𝟐 +𝒑𝒙+𝒒 𝟐 +…+ 𝑩 𝒏 𝒙+ 𝑪 𝒏 𝒙 𝟐 +𝒑𝒙+𝒒 𝒏

6. In general, we need to factor the denominator first
In general, we need to factor the denominator first. What happens if we don’t get linear factors?
If we get a repeated linear factor 𝒙−𝒓 𝒎 , then we’ll have corresponding partial fractions: 𝑨 𝟏 𝒙−𝒓 + 𝑨 𝟐 𝒙−𝒓 𝟐 +…+ 𝑨 𝒎 𝒙−𝒓 𝒎
If we get an irreducible quadratic factor 𝒙 𝟐 +𝒑𝒙+𝒒, then we’ll have a corresponding partial fraction: 𝑨𝒙+𝑩 𝒙 𝟐 +𝒑𝒙+𝒒
If we get a repeated irreducible quadratic factor 𝒙 𝟐 +𝒑𝒙+𝒒 𝒏 , then we’ll have corresponding partial fractions: 𝑩 𝟏 𝒙+ 𝑪 𝟏 𝒙 𝟐 +𝒑𝒙+𝒒 + 𝑩 𝟐 𝒙+ 𝑪 𝟐 𝒙 𝟐 +𝒑𝒙+𝒒 𝟐 +…+ 𝑩 𝒏 𝒙+ 𝑪 𝒏 𝒙 𝟐 +𝒑𝒙+𝒒 𝒏

7. In general, we need to factor the denominator first
In general, we need to factor the denominator first. What happens if we don’t get linear factors?
If we get a repeated linear factor 𝒙−𝒓 𝒎 , then we’ll have corresponding partial fractions: 𝑨 𝟏 𝒙−𝒓 + 𝑨 𝟐 𝒙−𝒓 𝟐 +…+ 𝑨 𝒎 𝒙−𝒓 𝒎
If we get an irreducible quadratic factor 𝒙 𝟐 +𝒑𝒙+𝒒, then we’ll have a corresponding partial fraction: 𝑨𝒙+𝑩 𝒙 𝟐 +𝒑𝒙+𝒒
If we get a repeated irreducible quadratic factor 𝒙 𝟐 +𝒑𝒙+𝒒 𝒏 , then we’ll have corresponding partial fractions: 𝑩 𝟏 𝒙+ 𝑪 𝟏 𝒙 𝟐 +𝒑𝒙+𝒒 + 𝑩 𝟐 𝒙+ 𝑪 𝟐 𝒙 𝟐 +𝒑𝒙+𝒒 𝟐 +…+ 𝑩 𝒏 𝒙+ 𝑪 𝒏 𝒙 𝟐 +𝒑𝒙+𝒒 𝒏

8. In general, we need to factor the denominator first
In general, we need to factor the denominator first. What happens if we don’t get linear factors?
If we get a repeated linear factor 𝒙−𝒓 𝒎 , then we’ll have corresponding partial fractions: 𝑨 𝟏 𝒙−𝒓 + 𝑨 𝟐 𝒙−𝒓 𝟐 +…+ 𝑨 𝒎 𝒙−𝒓 𝒎
If we get an irreducible quadratic factor 𝒙 𝟐 +𝒑𝒙+𝒒, then we’ll have a corresponding partial fraction: 𝑨𝒙+𝑩 𝒙 𝟐 +𝒑𝒙+𝒒
If we get a repeated irreducible quadratic factor 𝒙 𝟐 +𝒑𝒙+𝒒 𝒏 , then we’ll have corresponding partial fractions: 𝑩 𝟏 𝒙+ 𝑪 𝟏 𝒙 𝟐 +𝒑𝒙+𝒒 + 𝑩 𝟐 𝒙+ 𝑪 𝟐 𝒙 𝟐 +𝒑𝒙+𝒒 𝟐 +…+ 𝑩 𝒏 𝒙+ 𝑪 𝒏 𝒙 𝟐 +𝒑𝒙+𝒒 𝒏

9. Ex 𝑥+7 𝑥 𝑑𝑥

10. Note: If degree of top polynomial is ____ degree of bottom polynomial, then ______ first. Ex 4. 2 𝑥 3 −4 𝑥 2 −𝑥−3 𝑥 2 −2𝑥−3 𝑑𝑥

11. Note: If degree of top polynomial is ____ degree of bottom polynomial, then ______ first. Ex 4. 2 𝑥 3 −4 𝑥 2 −𝑥−3 𝑥 2 −2𝑥−3 𝑑𝑥 ≥

12. Note: If degree of top polynomial is ____ degree of bottom polynomial, then ______ first. Ex 4. 2 𝑥 3 −4 𝑥 2 −𝑥−3 𝑥 2 −2𝑥−3 𝑑𝑥 ≥
divide

13. Note: If degree of top polynomial is ____ degree of bottom polynomial, then ______ first. Ex 4. 2 𝑥 3 −4 𝑥 2 −𝑥−3 𝑥 2 −2𝑥−3 𝑑𝑥 ≥
divide

14. Ex −2𝑥+4 𝑥 𝑥−1 2 𝑑𝑥