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Def. 1. Rational function (rationas fraction) is called a function which can be represented as a ratio of two polynomials

. ( 1 )

Def. 2. Rational function (rationas fraction) (1) is called proper one if and improper otherwise ( ).

Theorem 1 (extraction of integer part of an improper rational function).Every improper rational function can be represented as a sum of some polynomial (so-called integer part) and a proper rational function.

■Let . Dividing the numerator by the denominator we get

where polynomials are a quotient and a remainder respectively and

is a proper rational function.■

Ex. 1. Extract an integer part of an improper rational fraction

a) The first (theoretical) way. After division of by we get

b) The second way. Subtracting and adding 1 in the numerator we’ll have

There are partial [simplest, elementary] fractions of 1- 4 types.

1. ;

2. ;

3. ;

4. .

We’ve integrated the fractions 1, 3 in the Point 2 of preceding lecture (ex. 14, 16, formulas (7), (10) and (11)). To integrate the fraction 2 we can put . Integration of the fraction 4 with the help of the substitution

leads to a linear combination of a simple integral

and the next one


As to evaluation of this latter see textbooks. For small values of k ( ) one can use the next substitution: .

Ex. 2.


Thus we can say that we able to integrate partial fractions 1 – 4.

Theorem 2(a partial decomposition of a proper rational function).Every proper rational function can be represented as a linear combination of partial fractions.

For example

Here are some unknown numbers (undetermined coefficients), which one can find by so-called method of undetermined coefficients.

Corollary.Every rational function can be integrated by virtue of the linear property of indefinite integral.

Rule of integration of a rational function.To integrate a rational function it’s necessary:

1. To extract an integer part of the function if it is improper one.

2. To factorize the denominator of obtained proper function into a product of polynomials of degree not higher than two.

3. To make a partial decomposition of the proper function.

4. To integrate all the terms of the obtained algebraic sum.

5. To write an answer.

Ex. 3.


Ex. 4. Evaluate the indefinite integral .

1 step (factorizing the denominator of the proper rational function).


2 step (a partial decomposition of the proper function).

Let’s assign two particular values to x in (*), namely .

3 step (integration of the given function by integration of its partial decomposition).

Ex. 5. Calculate the next indefinite integral

1 step (a partial decomposition of the proper rational function with factorized denominator).


. (**)

Assigning tree arbitrary values to x in (**), for example , we get a system of linear equations in ,

2 step (integration of all the terms of obtained partial decomposition of the integrand)




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