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
Calculus – How Do We Know the Integral of sin(x)?
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Best Answer
This is an interesting question and I understand the broader implications , but I will focus on the statement that computing a (definite) integral using the basic definition as a limit of Riemann sums is intractable for all but the simplest functions.
Granted, computation via the Fundamental Theorem of Calculus, is often the most expedient approach, but there comes a point where finding the anti-derivative in terms of elementary functions also is intractable. Furthermore, the bar for computation via the basic definition is perhaps not as high as you seem to think.
Presumably in your exercise you computed something like
$$\int_0^1 t^2 dt = \lim_{n \to \infty}\frac{1}{n} \sum_{k=1}^n\left( \frac{k}{n}\right)^2 = \frac{1}{3}, $$ or, even more generally, $$\int_0^x t^2 dt = \lim_{n \to \infty}\frac{1}{n} \sum_{k=1}^n\left( \frac{kx}{n}\right) = \frac{x^3}{3}, $$
and this was facilitated by knowing
$$\sum_{k=1}^n k^2 = \frac{n(n+1)(2n+1)}{6}.$$
Now consider your example, $\sin x$. I would assume you are aware of such basic properties as $\cos 2x = 1 - 2 \sin^2 x$ and $\lim_{x \to 0} \sin x / x = 1.$ Possibly less apparent is
$$\tag{1}\sum_{k=1}^n \sin (ky) = \frac{\sin \left(\frac{ny}{2} \right) \sin\left(\frac{(n+1)y}{2} \right)}{\sin\left(\frac{y}{2} \right)}.$$
This identity can be derived a number of ways, one being taking the imaginary part of the geometric sum $\sum_{k=1}^n (e^{iy})^k.$ As in your exercise where you knew the closed form for the sum of the squares, you can use $(1)$ to compute
$$\int_0^x \sin t \, dt = \lim_{n \to \infty}\sum_{k=1}^n\sin \left(\frac{kx}{n} \right)\left(\frac{kx}{n} - \frac{(k-1)x}{n} \right) = \lim_{n \to \infty}\frac{x}{n}\sum_{k=1}^n\sin \left(\frac{kx}{n} \right).$$
Using $(1)$ with $y = x/n$, we have
$$\begin{align}\frac{x}{n}\sum_{k=1}^n\sin \left(\frac{kx}{n} \right) &= \frac{x}{n}\frac{\sin \left(\frac{x}{2} \right) \sin\left(\frac{x}{2} + \frac{x}{2n} \right)}{\sin\left(\frac{x}{2n} \right)} \\ &= \frac{x}{n}\frac{\sin \left(\frac{x}{2} \right) \left[\sin\left(\frac{x}{2}\right)\cos\left(\frac{x}{2n}\right)+ \sin\left(\frac{x}{2n}\right) \cos\left(\frac{x}{2}\right)\right]}{\sin\left(\frac{x}{2n} \right)} \\ &= \frac{x\sin \left(\frac{x}{2} \right) \cos \left(\frac{x}{2} \right) }{n} + \frac{2\sin^2 \left(\frac{x}{2} \right) \cos\left(\frac{x}{2n}\right) }{\sin\left(\frac{x}{2n} \right)/ \frac{x}{2n} }\end{align}.$$
Now if we take the limit as $n \to \infty $ we see $ \frac{x}{2n} \to 0$ and
$$\int_0^x \sin t \, dt = \lim_{n \to \infty}\frac{x}{n}\sum_{k=1}^n\sin \left(\frac{kx}{n} \right) = 2\sin^2 \left(\frac{x}{2}\right) = 1 - \cos x = \cos 0 - \cos x.$$