Are you really sure this integral is trivial??

Hi! Today we are going to calculate the integral of an elementary function, cos(x), although it might seem trivial for people who are already familiar with Calculus, this approach might not be as easy as you think.

Really all we are doing is applying the Riemann definition of the integral, as it's done many times in Calculus courses, but this time we are applying it to an indefinite integral, meaning that we will be able to obtain the general antiderivative, and see a more explicit form of the famous constant of integration C.

The sum that we have to solve for this integral can be tricky if one doesn't get the right idea, so here are some sites and articles about this particular kind of sum, which is actually really useful in other fields:
Michael P. Knapp: https://www.maa.org/sites/default/files/Knapp200941575.pdf
Matthew Brett: https://matthew-brett.github.io/teaching/sums_of_cosines.html

The video has been quite long because I wanted to make clear what definitions and formulas we were going to use, so, if you just want to go to the most interesting part of the video, where the integral is being solved, here are the tiem stamps:
0:00 Introduction
1:30 Riemann Integral
5:25 Indefinite Integrals
10:30 Calculating the Integral
11:28 1. Sines Sum from scratch
21:55 2. Taking the Limit
29:50 3. Commentary on the division of sin(Delta x/2) (The base of the rectangles has to be smaller that 2pi!)
36:20 As we would expect, Barrow's Law gives us the same result!
37:20 Interpretation of the Indefinite Integral: Integrals make the values of areas equal to the values of lengths!

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Видео Are you really sure this integral is trivial?? канала Sandro’s Space