Ostrogradski method

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

$$ \tag{1 } Q( x) = ( x - a _ {1} ) ^ {\alpha _ {1} } \cdots ( x - a _ {r} ) ^ {\alpha _ {r} } ( x ^ {2} + p _ {1} x + q _ {1} ) ^ {\beta _ {1} } \cdots ( x ^ {2} + p _ {s} x + q _ {s} ) ^ {\beta _ {s} } , $$

where $ a _ {i} , p _ {j} , q _ {j} $ are real numbers, $ ( p _ {j} ^ {2} /4)- q _ {j} < 0 $, $ \alpha _ {i} $ and $ \beta _ {j} $ are natural numbers, $ i = 1, \dots, r $, $ j = 1, \dots, s $, and let

$$ \tag{2 } \left . \begin{array}{c} Q _ {1} ( x) = ( x - a _ {1} ) ^ {\alpha _ {1} - 1 } \cdots ( x - a _ {r} ) ^ {\alpha _ {r} - 1 } ( x ^ {2} + p _ {1} x + q _ {1} ) ^ {\beta _ {1} - 1 } \cdots ( x ^ {2} + p _ {s} x + q _ {s} ) ^ {\beta _ {s} - 1 } , \\ Q _ {2} ( x) = ( x - a _ {1} ) \cdots ( x - a _ {r} ) ( x ^ {2} + p _ {1} x + q _ {1} ) \cdots ( x ^ {2} + p _ {s} x + q _ {s} ). \end{array} \right \} $$

Then real polynomials $ P _ {1} ( x) $ and $ P _ {2} ( x) $ exist, the degrees of which are respectively less than the degrees $ n _ {1} $ and $ n _ {2} = r + 2s $ of the polynomials $ Q _ {1} ( x) $ and $ Q _ {2} ( x) $, such that

$$ \tag{3 } \int\limits \frac{P(x)}{Q(x)} dx = \ \frac{P _ {1} ( x) }{Q _ {1} ( x) } + \int\limits \frac{P _ {2} ( x) }{Q _ {2} ( x) } dx. $$

It is important that the polynomials $ Q _ {1} ( x) $ and $ Q _ {2} ( x) $ can be found without knowing the decomposition (1) of the polynomial $ Q( x) $ into irreducible factors: The polynomial $ Q _ {1} ( x) $ is the greatest common divisor of the polynomial $ Q( x) $ and its derivative $ Q ^ \prime ( x) $ and can be obtained using the Euclidean algorithm, while $ Q _ {2} ( x) = Q( x)/Q _ {1} ( x) $. The coefficients of the polynomials $ P _ {1} ( x) $ and $ P _ {2} ( x) $ 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 $ P _ {1} ( x)/Q _ {1} ( x) $ in formula (3) is the algebraic part of the indefinite integral $ \int P( x)/Q( x) dx $.

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