Chapter 7 – Techniques of Integration

1. Chapter 7 – Techniques of Integration 7.4 Integration of Rational Functions by Partial Fractions 7.4 Integration of Rational Functions by Partial Fractions

2. When do we use it? • In Calculus, there are several procedures that are much easier if we can take a rather large fraction and break it up into pieces. The procedure that can decompose larger fractions is called Partial Fraction Decomposition (PFD). We will proceed as if we are working backwards through an addition of fractions with LCD. 7.4 Integration of Rational Functions by Partial Fractions

3. Rational Functions • Rational functions are functions of the form Where P(x) and Q(x) are polynomials. • Example: 7.4 Integration of Rational Functions by Partial Fractions

4. Long Division • How do we perform the following integration? • Whenever the degree of the Numerator is greater than the degree of the denominator then we perform long divisionand then integrate. 7.4 Integration of Rational Functions by Partial Fractions

5. Partial Fractions • Whenever the degree of the Denominator is greater than the degree of the Numerator then we split the fraction into the sum of fractions. • Ex: 7.4 Integration of Rational Functions by Partial Fractions

6. Different cases of PFD • The partial fraction decomposition of a rational function R = P/Q, with deg(P) < deg(Q), depends on the factors of the denominator Q. It may have following types of factors: • Simple, non-repeated linear factors ax + b. • Repeated linear factors of the form (ax + b)k, k > 1. • Simple, non-repeated quadratic factors of the type ax2 + bx + c. Since we assume that these factors cannot be factorized anymore, we have b2 – 4 ac < 0. • Repeated quadratic factors (ax2 + bx + c)k, k>1. Also in this case we have b2 – 4 ac < 0. • We will consider each of these four cases separately. 7.4 Integration of Rational Functions by Partial Fractions

7. Different Cases of PFD • Distinct Linear Factors • Repeated Linear Factors • Distinct Irreducible Quadratic Factors • Repeated Irreducible Quadratic Factors 7.4 Integration of Rational Functions by Partial Fractions

8. Useful Integration Formulas • Sometimes, after we go through the steps of partial fractions decomposition, an integral can be evaluated using the formulas below: • Now let’s examine the cases more thoroughly. 7.4 Integration of Rational Functions by Partial Fractions

9. Case I: Distinct Linear Factors • This means we can write the denominator as where no factor is repeated and no factor is a constant multiple of another. In this case, there exists constants A1, A2, …, A3 such that 7.4 Integration of Rational Functions by Partial Fractions

10. To get the equations for A and B we use the fact that two polynomials are the same if and only if their coefficients are the same. Example 1 • By the result concerning Case I, we can find numbers A and B such that • Now we want to find A and B. 7.4 Integration of Rational Functions by Partial Fractions

11. Example 1 continued • Now we can rewrite the fraction and then integrate. 7.4 Integration of Rational Functions by Partial Fractions

12. Case II: Repeated Linear Factors • This means we can write the denominator as where a factor is repeated r times. In this case, there exists constants A1, A2, …, A3 such that 7.4 Integration of Rational Functions by Partial Fractions

13. Example 2 • The denominator of the rational function above can be factored as follows: • We can then use the partial fraction as follows • Now we need to find A, B, and C. 7.4 Integration of Rational Functions by Partial Fractions

14. Example 2 continued • Now we are ready to integrate. 7.4 Integration of Rational Functions by Partial Fractions

15. Case III: Distinct Irreducible Quadratic Factors • This means the denominator contains a factor where b2 – 4ac < 0 and no factor is repeated or is a constant multiple of another. In this case, there exists constants A1, A2, …, A3 such that 7.4 Integration of Rational Functions by Partial Fractions

16. Example 3 • The denominator of the rational function above can be factored as follows: • We can then use the partial fraction as follows • Now we need to find A, B, and C. 7.4 Integration of Rational Functions by Partial Fractions

17. Example 3 continued • Now we are ready to integrate. 7.4 Integration of Rational Functions by Partial Fractions

18. Case IV: Repeated Irreducible Quadratic Factors • This means the denominator contains a factor where b2 – 4ac < 0 and a factor is repeated r times. In this case, there exists constants A1, A2, …, A3 such that 7.4 Integration of Rational Functions by Partial Fractions

19. Example 4 • Write out the form of the partial fraction decomposition of the function 7.4 Integration of Rational Functions by Partial Fractions

20. Example 5 • We can simplify the function to be integrated by performing polynomial division first. This needs to be done whenever possible. We get: • Partial fraction decomposition for the remaining rational expression leads to • Now we can integrate 7.4 Integration of Rational Functions by Partial Fractions