Expansion of the proper rational fraction into partial fractions

Integration of the partial rational fractions.

Example 7.2.

; ; - partial fractions.

I. ;

II. , where

.

III. .

As a result of this substitution considered integral leads to the sum of table integrals

and .

IV. .

Integrals in the form IV can be evaluated by substitution which was used for integrals in the form III.

Let - proper irreducible rational fraction. Consider two cases:

Case 1. has only real roots multiplicity of , so

then

=

+ (7.1), were - coefficients must be found.

Case 2. except real roots multiplicity of , has roots which are solutions of equation when , so

then

=+

+ (7.2) were - coefficients must be found.

Example 7.3. Decompose rational fraction into the sum of partial fractions.

a)

Solution:

Using expansion (7.1) we have

, where - unknown coefficients.

To find unknown coefficients we reduce the fractions of the right side of last equation to the common denominator and equate numerators of both sides of the equation.

Equate the coefficient of corresponding powers of, and solve the resulting equations for the unknown coefficients

;

.

Дата добавления: 2014-01-11