How does the fundamental theorem of algebra help with integrating rational functions?

The fundamental theorem of algebra is essential for integrating rational functions because it guarantees that any polynomial with real coefficients can be factored into a product of linear and quadratic factors.1

In the context of calculus, this property allows for the method of partial fraction decomposition. By factoring the denominator of a rational function into these linear and quadratic components, a complex rational expression can be broken down into a sum of simpler, elementary fractions. These simpler terms are then straightforward to integrate using standard techniques, such as those involving logarithms or inverse trigonometric functions. Without the assurance provided by the theorem that such a factorization is always possible, the systematic integration of arbitrary rational functions would not be mathematically guaranteed.1