TITLE: How do we know what the integral of $\sin (x)$ is?
QUESTION [8 upvotes]: Since I started more or less formally learning the foundations of calculus, I naturally came across the Riemann definition of the integral. At first I considered this definition intuitive but not really useful since to obtain a closed form expression, one needed to add an infinite amount of values. Later on, an exercise prompted me to calculate a Riemann Integral, and from the definition and the expression for the sum of squares, I was able to calculate the limit with nothing more than I had learned at school.

This was a revelation for me, since so far I had considered the definition a mere formalism. Now I knew how it gave results. The next integral I tried to calculate this way was, for obvious reasons $\sqrt {1-x^2}$. Unfortunately, I found the sum intractable and gave up.

I started to question the usefulness of the definition again. If it only works for simple functions like polynomials, how did we ever find out that the integral of $\sin (x)$ is $-\cos (x)$? Did we use the Riemann definition or did we just say "the derivative of $-\cos (x)$ is $\sin (x)$ and therefore its integral must be $-\cos (x)$?

I would like to get some insight into the theory as well as the history that led to the tables of integrals we have today

REPLY [3 votes]: Expanding the comments of Doug M and benguin: this is a very simplified version of the history.
Gregory/Barrow/Newton proved (more or less) the Fundamental Theorem of Calculus with integral = area under the curve.
About the formulas $\sin' = \cos$, $\cos' = -\sin$, is really difficult to say who proved first. Maybe Roger Cotes? See The calculus of the trigonometric functions for details and also How were derivatives of trigonometric functions first discovered?
Also very interesting: Some History of the Calculus of the
Trigonometric Functions includes the proof by Archimedes of our formula
$$\int_0^\pi\sin = 2$$
than can be easily generalized (Archimedes dont't do this) to
$$\int_0^\alpha\sin x\,dx = 1 - \cos\alpha.$$
The section about Pascal is equally interesting.