Some examples.

Procedure:

The method of Partial Fractions provides a way to integrate all rational functions. Recall that a rational function is a function of the form

where P and Q are polynomials. Hence, we want a technique to find the integral

1. The technique requires that the degree of the numerator be less than the degree of the denominator. If this is not the case then we first must divide the numerator into the denominator.

   Example 1.

2. We factor the denominator Q into powers of distinct linear terms and powers of distinct quadratic polynomials which do not have real roots.

   More information on factoring.

3. If r is a real root of order k of Q, then the partial fraction expansion of P/Q contains a term of the form
   

   where A₁, A₂, ..., A_k are unknown constants.

4. If Q has a quadratic factor ax² + bx + c which corresponds to a complex root of order k, then the partial fraction expansion of P/Q contains a term of the form
   

   where B₁, B₂, ..., B_k and C₁, C₂, ..., C_k are unknown constants.

5. After determining the partial fraction expansion of P/Q, we set P/Q equal to the sum of the terms of the partial fraction expansion.

   Example 2.

6. We then multiply both sides by Q to get some expression which is equal to P.

7. Now, we use the property that two polynomials are equal if and only if the corresponding coefficients are equal. If P is a polynomial of degree n, then this gives us n +1 equations in n +1 unknowns: the A_i's, the B_i's and the C_i's.

   Example 2 (cont.).

8. We can use a graphing calculator to solve for the A_i's, the B_i's and the C_i's.

   Example 2 (cont.).

9. We express the integral of P/Q as the sum of the integrals of the terms of the partial fraction expansion.

   Example 2 (cont.).

10. Integrate linear factors:

    • 
      Discussion.

    • for n > 1.
      Discussion.

11. Integrate quadratic factors:

    1. Some simple formulas:

       • 
         Discussion.

       • 

    2. General formulas.

    Example 2 (cont.).

Some drill problems
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Some more drill problems.