Before the development of special functions, antiderivatives were solved using techniques such as integration by parts, partial fractions, and clever substitutions. These methods are still commonly used in Calculus II and beyond to solve antiderivatives of ordinary, analytic functions.
One approach to solving antiderivatives beyond what is typically learned in Calculus II is to use series expansions. This involves expressing the function as a series of simpler functions and integrating each term individually. This method can be particularly useful for functions that cannot be easily integrated using other techniques.
Another method is to use numerical integration techniques, such as the trapezoidal rule or Simpson's rule. These methods involve approximating the area under the curve using smaller intervals and summing the areas of each interval. While these methods may not provide an exact solution, they can provide a close approximation.
In addition, integral tables were also used extensively before the development of special functions. These tables contained pre-calculated values for commonly encountered integrals, allowing mathematicians to look up the solution for a given function.
Overall, solving antiderivatives before special functions required a combination of mathematical techniques, such as integration by parts and substitution, as well as the use of pre-calculated values in integral tables. With the development of special functions, many of these integrals can now be solved using specific formulas, making the process more efficient and less reliant on manual techniques.
