TITLE: Integral calculus analog to the difference quotient
QUESTION [5 upvotes]: In differential calculus, everyone learns the classic limit definition of the derivative, also known as the difference quotient. Namely, $$f'(x)= \lim_{h\to0} \frac{f(x+h)-f(x)}{h}$$
My question is, is there an analog to the difference quotient in integral calculus?
I did some research and found this, which works for all $f :[a,b] \to \Bbb R$ (good enough for me): 
$$\int_{a}^{b}f(x)dx=\lim_{n\to{\infty}}\sum_{k=1}^{n}f\left(a+k\frac{b-a}{n}\right)\frac{b-a}{n}$$
However, to someone that has yet to learn how to tackle infinite sums as involved as these, this definition is far from ideal. 
Is there a "simple" formula like the difference quotient for finding either integrals (or antiderivatives)? By simple, I mean something in terms of the original function, like the difference quotient.
I understand that the integral is fundamentally different from the derivative as it is a sum, but the Riemann sum seems complicated at this point. Can this sum be reduced into a simpler "difference quotient like" general form?

REPLY [2 votes]: Quite simply: no, there isn't.
Beyond the more obvious differences, antiderivatives are just more difficult than derivatives, in a qualitative way. Derivatives can be computed systematically, for probably any function you could come up with. It's not even very hard: a good project for first-year computer science students is to make a program that analytically computes derivatives, like taking in sin(cos(x)+x^2) and spitting out (2*x-sin(x))*cos(x^2+cos(x)).
For antiderivatives, it's a whole nother story. There are many fairly straightforward functions that can't have their antiderivative expressed in terms of your basic functions. One classic example is the logarithmic integral,
$$ \int \frac{1}{\ln x}\,dx = \mathrm{li}(x) $$
You can't express $\int \frac{1}{\ln x}\,dx$ in terms of any combination of adding, multiplying, constants, dividing, powers, roots, exponentials, logarithms, and trig functions. That is to say, it isn't considered an elementary function. Other integrals that aren't elementary functions are $\int \sin(x^2)\,dx,\quad$ $\int \sqrt{1-x^4}\,dx,\quad$ and $\int \exp(-x^2)\,dx$. The impossibility of this task has actually been rigorously proven.
(Unfortunately, all those integrals are also pretty important: they come up, respectively, in the fields of number theory, optics, modelling the motion of a pendulum without a small angle approximation, and statistics. So, we've given them unique names, like li(x), S(x), K(x), and erf(x).)
In some sense, if there was an extremely simple definition of the antiderivative, like there is for the derivative, we would expect that it would lead us to a much simpler way to find antiderivatives. The fact that so many integrals can't be solved in any kind of straightforward manner is evidence that there won't be a simple definition.