Question: Hi all, first, I'd like to apologize if the term "singularity" is being misused. I have the following integral:

$\int \_{0}^{\pi/2} \sqrt{ r \left( x \right) ^{2}
+ \left( {\frac {d}{dx}} r \left( x \right) \right) ^{2}} \left( \ln \left( {
\frac {\sin \left( x \right) }{1-\cos \left( x \right) }} \right) + 1 \right) {dx}$

Note that the integrand shoots to infinity as x approaches 0. It's not too hard to establish that the integral will nonetheless always converge, regardless of the choice of r. My question is: will an extremizer satisfy the Euler-Lagrange Equation on the interval (0,Pi/2], and how can I tell? Sorry if this is trivial, I do not know much about the calculus of variations.

Answer: 
There are very general triangulability results for real (semi)algebraic sets (sets cut out by inequalities of real polynomials), and even for semianalytic and subanalytic sets. Lojasiewicz has some papers from the 60s on semianalytic sets; Hironaka and Hardt also have papers on the subject; and the book Real Algebraic Geometry by Bochnak, Coste, and Roy treats at least the semialgebraic case.
Actually, any complex algebraic variety can be triangulated as a **pseudomanifold**, which is a simplicial complex such that, if the variety has real dimension $n$, then any simplex is contained in an $n$-simplex, any $(n-1)$-simplex is contained in exactly two $n$-simplexes, and any two $n$-simplexes can be connected by a sequence of $n$-simplexes such that two consecutive ones share exactly one $(n-1)$-face.