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