It is often the case, when evaluating definite integrals, that an antiderivative for the integrand cannot be found, or is extremely difficult to find. In some instances, a numerical approximation to the value of the definite value will suffice. The following techniques can be used, and are listed in rough order of ascending complexity.
A common technique of approximating common trigonometric functions is to use the Taylor-Maclaurin series. Term-by-term integration allows one to easy compute the value of the integral by hand, well up to 5 decimal places of precision, and up to 10 given a factorial table.
For example, using the Maclaurin series of , one can easily approximate its integral with a polynomial.
We can then easily integrate each term, taking and to be constants.
We can easily find the constant term by inspecting the known principle integral, , and the new series. This nets us the final equation.
While this is a rather fast-converging series, converging at digits of significance, it is relatively useless, since factorials are expensive to compute.