My favorite theorem is the Fundamental theorem of Calculus. The second part to be more precise.
It states that if you have a closed interval, [a,b] and a function f who's antiderivative is F, then
- $ \int_a^b f(x)\,dx = F(b) - F(a)\,. $
I really like this theorem, because it made a lot of sense to me when I learned it. For years I had been studying all sorts of seemingly random math topics, and calculus really brought all of it together and made it "click" in my head. And the fundamental theorem is the best representation of my fondness for calculus as a whole.
