I have to deal with this integral in order to compute the period of a pendulum
$ \int^{\theta_{0}}_{0}\frac{d\theta}{\sqrt{\cos\theta_{0}-\cos\theta}} $
I was asked by my instructor to solve this with a taylor expansion for cos up to $O(\theta^4)$ I plugged in
$ \cos\theta_{0} = 1 - \frac{\theta_{0}^2}{2!} + \frac{\theta_{0}^4}{4!} $ $ \cos\theta = 1 - \frac{\theta^2}{2!} + \frac{\theta^4}{4!} $
$ \int^{\theta_{0}}_{0}\frac{d\theta}{\sqrt{\frac{\theta^2}{2}-\frac{\theta_0^2}{2}-\frac{\theta^4}{4!}+\frac{\theta_0^4}{4!} }} $
but the following integral eluded simplication ( I spent alot of time here). Later, I was able to solve this problem by using the substitution $\cos\theta = 1-2\sin^2\frac{\theta}{2}$ the integral is then solvable by series in terms of a binomial expansion in terms of $k^2x^2$ where $\sin x = \frac{\sin\frac{\theta}{2}}{\sin\frac{\theta_0}{2}}$
However, my task was not to do the expansion of a binomial but rather to solve the integral with an expansion for cos. Thus i am still lost as to how to proceed.