# Is the integral of a rational function always a rational function?

## Question:

Is the integral of a rational function always a rational function?

## Antiderivatives; Rational Functions:

To find antiderivatives (indefinite integrals) of rational functions, we can use the partial fraction decomposition to facilitate the process by finding the antiderivatives of each rational term, which are simpler than the original rational function. The following two examples of irreducible fractions are important in this regard:

{eq}\frac{1}{x+\alpha} \text{ for any real }\alpha, {/eq}

and

{eq}\frac{1}{\alpha^2+x^2}\text{ for }\alpha\ne 0. {/eq}

These two types of irreducible fractions can be used to answer this problem.

The following two examples of irreducible fractions appear frequently when using the method of partial fraction decomposition to facilitate the...

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