TITLE: Is there a definite integral for which the Riemann sum can be calculated but for which there is no closed-form antiderivative?
QUESTION [15 upvotes]: Some definite integrals, such as $\int_0^\infty e^{-x^2}\,dx$, are known despite the fact that there is no closed-form antiderivative. However, the method I know of calculating this particular integral (square it, and integrate over the first quadrant in polar coordinates) is not dependent on the Riemann sum definition. What I thought might be interesting is a definite integral $\int_a^bf(x)\,dx$ for which the limit of the Riemann sums happens to be calculable, but for which no closed-form antiderivative of $f$ exists. Of course there are some obvious uninteresting examples, like integrating odd functions over symmetric intervals, but one doesn't need Riemann sums to calculate these uninteresting examples.
Edit: To make this a bit clearer, it would be nice to have a "natural" continuous function $f(x)$ where by some miracle $\lim_{n\to\infty} \sum_{i=1}^nf(x_i)\Delta x$ is computable (for some interval $[a,b]$) using series trickery, but for which no antiderivative exists composed of elementary functions.

REPLY [4 votes]: This may not answer your question, but I read of another example of an integral which can be calculated without finding an anti-derivative in Halmos's Problems for Mathematicians, Young and Old is the integral:
$$ \int_{0}^{1} \int_{0}^{1} \frac{1}{1-xy} dx dy.$$
The trick here is to look at the integrand at each $x,y$ as a number $\frac{1}{1-r}$ where $0<r=xy<1$ and follow your heart. 

Edit:
On second thought, this answer is probably not to the point, since the solution will require a convergence theorem (monotone convergence works), and then we need an anti-derivative-less calculation of integrals of the form $\int_{0}^{1} x^k dx$. There are such (found here, for instance). In any case, the method for finding this integral invites the generalizations$$\int_{0}^{1} \cdots \int_{0}^{1} \int_{0}^{1} \frac{1}{1- x_1 x_2 \cdots x_n} dx_1 dx_2 \cdots dx_n$$
where $n>1$.