# Ostrogradski method

A method for isolating the algebraic part in indefinite integrals of rational functions. Let and be polynomials with real coefficients, let the degree of be less than the degree of , so that is a proper fraction, let

(1) |

where are real numbers, , and are natural numbers, , , and let

(2) |

Then real polynomials and exist, the degrees of which are respectively less than the degrees and of the polynomials and , such that

(3) |

It is important that the polynomials and can be found without knowing the decomposition (1) of the polynomial into irreducible factors: The polynomial is the greatest common divisor of the polynomial and its derivative and can be obtained using the Euclidean algorithm, while . The coefficients of the polynomials and can be calculated using the method of indefinite coefficients (cf. Undetermined coefficients, method of). The Ostrogradski method reduces the problem of the integration of a real rational fraction to the integration of a rational fraction whose denominator has only simple roots; the integral of such a fraction is expressed through transcendental functions: logarithms and arctangents. Consequently, the rational fraction in formula (3) is the algebraic part of the indefinite integral .

The method was first published in 1845 by M.V. Ostrogradski (see ).

#### References

[1a] | M.V. Ostrogradski, Bull. Sci. Acad. Sci. St. Petersburg , 4 : 10–11 (1845) pp. 145–167 |

[1b] | M.V. Ostrogradski, Bull. Sci. Acad. Sci. St. Petersburg , 4 : 18–19 (1845) pp. 286–300 |

**How to Cite This Entry:**

Ostrogradski method.

*Encyclopedia of Mathematics.*URL: http://encyclopediaofmath.org/index.php?title=Ostrogradski_method&oldid=18118