Many important integrands, such as $e^{-x^2}$ or $\sin(x)/x$, have no elementary antiderivative, so their definite integrals cannot be evaluated by the Fundamental Theorem of Calculus directly. Instead, one substitutes the Maclaurin series for the integrand and integrates term by term (valid within the radius of convergence), then truncates to a partial sum, using the Taylor remainder bound to control the resulting error.