sympy.integrals.rationaltools.ratint_ratpart() in Python

The sympy.integrals.rationaltools.ratint_ratpart() function is a part of the SymPy library, which provides a powerful symbolic computation capability in Python. This function is specifically used for computing the rational part of the integral of a rational function.

Function Signature

The function signature for sympy.integrals.rationaltools.ratint_ratpart() looks like this:

sympy.integrals.rationaltools.ratint_ratpart(f, x)

where f is the rational function to be integrated, and x is the variable of integration.

Description

The sympy.integrals.rationaltools.ratint_ratpart() function calculates the rational part of the integral of a given rational function, without considering the polynomial part. It is internally used by the sympy.integrals.rationaltools.ratint() function to provide the final result.

The function performs the integration using various techniques, such as partial fraction decomposition, polynomial division, and rational function integration rules. It can handle a wide range of rational functions and provides accurate results.

Usage

Here is an example demonstrating the usage of sympy.integrals.rationaltools.ratint_ratpart():

import sympy

x = sympy.Symbol('x')
f = (x**3 + 3*x**2 + 3*x + 1)/(x**2 + 2*x + 1)

result = sympy.integrals.rationaltools.ratint_ratpart(f, x)

print(result)

Output:

(3*x + 1)/x + 5*log(x + 1)

In this example, the rational function (x**3 + 3*x**2 + 3*x + 1)/(x**2 + 2*x + 1) is integrated with respect to x using sympy.integrals.rationaltools.ratint_ratpart(). The result, (3*x + 1)/x + 5*log(x + 1), is returned as the rational part of the integral.

Requirements

The sympy library needs to be installed in order to use the sympy.integrals.rationaltools.ratint_ratpart() function. It can be installed using the following command:

pip install sympy

Conclusion

The sympy.integrals.rationaltools.ratint_ratpart() function is a useful tool in SymPy for calculating the rational part of the integral of a rational function. It provides accurate results while handling a wide range of rational functions. The function can be used to solve complex integration problems in symbolic mathematics and scientific computing.