Difficult Integration functions that require Taylor Series?

sin(x^2) and cos(x^2) (or, if you treat the subject using complex numbers, these functions' related cousin e^(x^2)) appear in the study of diffraction (if you google "Fresnel integrals" you may find some useful links; I don't know the subject at all).

If you study the motion of a pendulum, in a first-year physics course one usually makes "small angle" approximations (that sin t is approximately t) to "linearize" the resulting equation and get nice solutions in closed form. Without any approximation, exact study of pendulum motion requires "special functions" (or series methods) because the integrals one needs to take cannot be done by elementary means. See the Wikipedia link below; in particular the section "arbitrary-amplitude period" for the definite integral expressing the period of a pendulum motion in terms of the length l of the pendulum, the gravitational constant g, and the angular displacement theta_0 at which you begin the periodic motion. The relevant function is f(t) = 1/sqrt(cos(t) - k) where k is some constant, and cannot be integrated by ordinary means.

I prefer this latter example myself. If you don't like pendulums you can give a different physical analogy: you have a spherical bowl and a tiny ball (so small and light that the effects of its rolling can be neglected, and of course friction doesn't exist). You place the ball at the lip of the bowl and let go. It turns out that you need to take "difficult" integrals to figure out something as simple as how long it takes for the ball to reach the bottom of the bowl!