It's not particularly famous, but it should be; something almost all mathematics students encounter without realizing it, the definite integration problem. The problem of whether a given indefinite integral has a closed form antiderivitive expressible in elementary functions is solved, in the form of a semi-algorithm, the Risch algorithm. There is no similar semi-algorithm or answer on the existence of an algorithm for definite integrals: Does a given definite integral have a solution expressible in elementary functions?
The existence of such an algorithm would, assuming it's efficient and terminating in a reasonable amount of time, answer many open problems in transcendence theory along with many applications in high performance computing and computer algebra systems.
Consider, for example, some of the highest voted questions on this site related to definite integrals, one with a closed form solution, and one without:
$$I=\int_{-1}^1\frac1x\sqrt{\frac{1+x}{1-x}}\ln\left(\frac{2\,x^2+2\,x+1}{2\,x^2-2\,x+1}\right)\ \mathrm dx.$$
This can be attacked with a few techniques for an exact answer here:
$\large\hspace{3in}I=4\,\pi\operatorname{arccot}$$\sqrt\phi$
But these techniques don't apply to every definite integral.
In particular this question asks for a closed form solution to this definite integral:
$$\int_{0}^{\Large\frac{\pi}{2}}\frac{1}{(1+x^2)(1+\tan x)}\,\mathrm dx$$
It's unknown what the closed form solution to this is, and it may be possible to nonconstructively prove it exists without actually being able to say what it is. The answer is ambiguous as to whether the particular closed form solution for this exists.
Now if these sorts of definite integrals had closed form antiderivitives, one could simply apply the fundamental theorem of calculus to give closed form solutions. But the Risch algorithm tells us closed form antiderivitives to these expressions don't exist so we must resort to heuristics that apply to specific classes of definite integrals. Some of them are quite exciting, like the countour integration approach used in the first problem; but this is far short of a generic algorithm analagous to Risch for definite integrals.
What's astounding to me is how little press this problem gets compared to big name problems of moderate interest and little applicable value like the Goldbach conjecture.
http://mathworld.wolfram.com/DefiniteIntegral.html