Integration of Rational Functions

The integration of rational functions refers to finding the antiderivative of a function that is the ratio of two polynomials. In general, these integrals do not have a straightforward formula and often require more advanced techniques such as polynomial division, partial fraction decomposition, or a substitution if it is an improper rational function (where the degree of the numerator is greater than or equal to the degree of the denominator), to reduce it to a form that can be integrated.

For the simple case of a linear rational function of the form ($a\times x+b$)/($c\times x+d$), we cannot provide a full solution through an arrow function since the process often involves conditional decisions based on the relationship between $a$, $b$, $c$, and $d$. However, we can format the integral for further analysis.

This type of integration is useful in many areas of calculus, physics, and engineering, particularly where the relationship between quantities can be expressed as a ratio of polynomials.