1. Partial Fraction Decompositions Rational Functions Partial Fraction Decompositions Finding Partial Fractions Decompositions Integrating Partial Fraction Decompositions Examples

2. Mika Seppälä: Partial Fraction Decompositions Rational Functions A function of the type P/Q, where both P and Q are polynomials, is a rational function. Definition Example The degree of the denominator of the above rational function is less than the degree of the numerator. Hence we may rewrite the above rational function in a simpler form by performing polynomial division. Rewriting For integration, it is always necessary to perform polynomial division first, if possible. To integrate the polynomial part is easy, and one can reduce the problem of integrating a general rational function to a problem of integrating a rational function whose denominator has degree greater than that of the numerator.

3. Mika Seppälä: Partial Fraction Decompositions Example of Integrating Rational Functions

4. Mika Seppälä: Partial Fraction Decompositions Partial Fraction Decomposition of the function to be integrated. Definition The above decomposition of the rational function (3x 2 +3x+2)/(x 3 +x 2 +x+1) is a partial fraction decomposition. Partial Fraction Decomposition is a rewriting of a rational function into a sum a rational functions with as simple denominators as possible. General partial fraction decomposition is technically complicated and involves several cases. It all starts with a factorization of the denominator. The type of the partial factor decomposition depends on the type of the factors of the denominator. The different cases will be explained on the following slides.

5. Mika Seppälä: Partial Fraction Decompositions Simple Linear Factors (1) Case I Partial Fraction Decomposition: Case I

6. Mika Seppälä: Partial Fraction Decompositions Simple Linear Factors (2) Example 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.

7. Mika Seppälä: Partial Fraction Decompositions Simple Quadratic Factors (1) Case II Partial Fraction Decomposition: Case II

8. Mika Seppälä: Partial Fraction Decompositions Simple Quadratic Factors (2) Example To get these equations use the fact that the coefficients of the two numerators must be the same.

9. Mika Seppälä: Partial Fraction Decompositions Repeated Linear Factors (1) Case III Partial Fraction Decomposition: Case III

10. Mika Seppälä: Partial Fraction Decompositions Repeated Linear Factors (2) Example Equate the coefficients of the numerators.

11. Mika Seppälä: Partial Fraction Decompositions Repeated Quadratic Factors (1) Case IV Partial Fraction Decomposition: Case IV

12. Mika Seppälä: Partial Fraction Decompositions Repeated Quadratic Factors (2) Example This can also be computed by Maple command convert.

13. Mika Seppälä: Partial Fraction Decompositions Integrating Partial Fraction Decompositions (1) Partial Fraction Decomposition is the main method to integrate rational functions. After a general partial fraction decomposition one has to deal with integrals of the following types. There are four cases. Two first cases are easy. Here K is the constant of integration.

14. Mika Seppälä: Partial Fraction Decompositions Integrating Partial Fraction Decompositions (2) The 3 rd case leads to two integrals in general. Here we have to compute integrals of the type: We start by the following rewriting Compute the first integral by a substitution. One gets:

15. Mika Seppälä: Partial Fraction Decompositions Integrating Partial Fraction Decompositions (3)

16. Mika Seppälä: Partial Fraction Decompositions Integrating Partial Fraction Decompositions (5) All these integrals can also be computed using the previous methods. Theorem

17. Mika Seppälä: Partial Fraction Decompositions Integration Algorithm Integration of a rational function f = P/Q, where P and Q are polynomials can be performed as follows. 1.If deg(Q)  deg(P), perform polynomial division and write P/Q = S + R/Q, where S and R are polynomials with deg(R) < deg(Q). Integrate the polynomial S. 2.Factorize the polynomials Q and R. Cancel the common factors. Perform Partial Fraction Decomposition to the simplified form of R/Q. 3.Integrate the Partial Fraction Decomposition.

18. Mika Seppälä: Partial Fraction Decompositions Examples (1)

19. Mika Seppälä: Partial Fraction Decompositions Examples (2)

20. Mika Seppälä: Partial Fraction Decompositions Examples (3) The last two integrals have to be computed by suitable substitutions.