Let's say that we want to evaluate without knowing the anti derivative of . How could we do this? To start let's consider the unit circle, we know the unit circle has arc-length . We can equivalently write the arc length of the unit circle as
Notice when , . So recalling that :
We can also use this to discover many other relations. For example
This was the method of Italian mathematician Giulio Fangnano.
We can of course immediately infer via the inverse function of the anti-derivative in question. We know via the fundamental theorem of calculus
And we know this is in the form . So we can infer , which leads us to the anti-derivative we assumed we didn't know at the beginning