PreCalculus students can use the Completing the Square Method to solve quadratic equations without the need to memorize the quadratic formula since this method naturally leads them to that formula. Calculus students, when studying integration, use various standard methods to compute integrals depending on the type of function to be integrated. Before integrating rational functions, students often need to know how to decompose the function by using the Partial Fraction Decomposition. In some cases, extending the Completing the Square Method beyond polynomial functions, to include rational functions, can be very helpful in avoiding lengthy computations where the potential of error is high. In this manuscript we propose an alternative method to the lengthy Partial Fraction Decomposition, used in standard calculus textbooks, to compute the indefinite integral of a family of rational functions. We will also demonstrate how the integral of the rational function, one over one plus x to the fourth power, can be thought of as a special case of the integral of the family of rational functions.