Scheme of integration of the rational fraction

1. Define the type of fraction: proper or improper.

2. If fraction is improper express the fraction as a sum of the polynomial and the proper fraction dividing the numerator by the denominator;

3. If fraction is proper using formulas (7.1) or (7.2) express the fraction as a sum of the partial fractions.

4. Find unknown coefficients of the expansion;

5. Evaluate partial integrals in the forms I-IV.

Example 7.4. Evaluate integrals: a), b)

a)

The integrand is the proper irreducible rational fraction. Using the expansion of the fraction into partial fractions from example (7.3), we have

=.

b)

The integrand is the improper rational fraction. We will divide the numerator by the denominator.

Decompose the improper rational fraction into the sum of the polynomial and proper fraction

=

;

=.

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