If you have a function that cannot be integrated but which has a fourth
derivative, you can approximate the definite integral to a high degree of
accuracy using Simpson's rule.

Choose \(\Delta x\) such that \([a,b]\) is divided into an even number of subintervals.

\[\int_{a}^b f(x)\,dx = \lim_{\Delta x \to 0^+} \frac{\Delta x}{3}[f(x_0) + 4f(x_1) + 2f(x_2) + 4f(x_3) + 2f(x_4) + ... + 4f(x_{n-1}) + f(x_n)] \]

For the error, find the maximum value \(M\) of \(f^{(4)}(x)\) on \([a,b]\).
\[\frac{b-a}{180}M(\Delta x)^4\]