Ostrogradsky's Method

Syntax:

[P_1,Q_1,P_2,Q_2] = ostrogradskysmethod(P,Q,x)

Description:

For an integral with an integrand that is a proper rational fraction, Ostrogradsky's decomposes the integral as

$\int \frac{P(x)}{Q(x)} \, dx = \frac{P_1(x)}{Q_1(x)} + \int \frac{P_2(x)}{Q_2(x)} \, dx. \tag*{} $

The inputs to ostrogradskysmethod are symbolic polynomials P and Q, with P being lesser degree than Q. The outputs are symbolic polynomials P_1, Q_1, P_2, and Q_2.

Examples:

Use Ostrogradsky's method to decompose an integral with P(x) = x^3-x^2+x+1 and Q(x) = (x^2+1)^3

syms x
P = x^3-x^2+x+1;
Q = (x^2+1)^3;
[P_1,Q_1,P_2,Q_2] = ostrogradskysmethod(P,Q,x)

P_1 =
x^3/4 - x^2/2 + (3*x)/4 - 1/2
Q_1 =
(x^2 + 1)^2
P_2 =
1/4
Q_2 =
x^2 + 1

Take the integral via Ostrogradsky's method and confirm that it matches MATLAB's solution

I = P_1/Q_1+int(P_2/Q_2)
I_c = int(P/Q)

I =
atan(x)/4 + (x^3/4 - x^2/2 + (3*x)/4 - 1/2)/(x^2 + 1)^2
I_c =
atan(x)/4 + (x^3/4 - x^2/2 + (3*x)/4 - 1/2)/(x^2 + 1)^2