A rational function

, where

*P*(*x*) and

*Q*(*x*)
are both polynomials, can be integrated in four steps:

- Reduce the fraction if it is improper (i.e. degree of
*P*(*x*) is greater than degree of *Q*(*x*));

- Factor
*Q*(*x*) into linear and/or quadratic (irreducible) factors;

- Decompose the fraction into a sum of partial fractions;

- Calculate integrals of each partial fraction.

Consider the specified steps in more details.

Step 1. Reducing an improper fraction

If the fraction is improper (i.e. degree of

*P*(*x*) is greater than degree of

*Q*(*x*)),
divide the numerator

*P*(*x*) by the denominator

*Q*(*x*) to obtain

where

is a proper fraction.

Step 2. Factoring *Q*(*x*) into linear and/or quadratic factors

Write the denominator

*Q*(*x*) as

where quadratic functions are irreducible, i.e. do not have real roots.

Step 3. Decomposing the rational fraction into a sum of partial fractions.

Write the function as follows:

The total number of undetermined coefficients

*A*_{i} , B_{i} , K_{i} , L_{i} , M_{i} , N_{i} , ...
must be equal to the degree of the denominator

*Q*(*x*).

Then equate the coefficients of equal powers of

*x* by multiplying both sides of the latter expression by

*Q*(*x*) and write the system of linear equations in

*A*_{i} , B_{i} , K_{i} , L_{i} , M_{i} , N_{i} , ....
The resulting system must always have a unique solution.

Step 4. Integrating partial fractions.

Use the following 6 formulas to evaluate integrals of partial fractions with linear and quadratic denominators:

For fractions with quadratic denominators, first complete the square:

where

Then use the formulas:

The integral

can be calculated in

*k* steps using the reduction formula: