Ostrogradski method
A method for isolating the algebraic part in indefinite integrals of rational functions. Let
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 |
Ostrogradski method. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Ostrogradski_method&oldid=52153