 This integral can be rewritten in terms of elliptic integrals as

where is the incomplete elliptic integral of the first kind defined by

Or more concisely by the substitution
expressing in terms of ,
Here is the complete elliptic integral of the first kind defined by

For comparison of the approximation to the full solution, consider the period of a pendulum of length 1 m on Earth ( = ) at initial angle 10 degrees is
The linear approximation gives

The difference between the two values, less than 0.2%, is much less than that caused by the variation of with geographical location.From here there are many ways to proceed to calculate the elliptic integral.Given and the Legendre polynomial solution for the elliptic integral