Can we always integrate numerically? I dont know if its suitable here or on Math SE, Most of the times, when I watched online lectures most lecturers say that if we cant solve a integral exactly we can always numerically integrate it. (E.g. in central force problem the orbit where we find $\theta(r)$ or $r(\theta)$ is pretty difficult to compute and professor said that if needed we can always integrate it numerically if we cannot do it exactly)
So is it always possible to numerically integrate? Is there any example in physics where it not possible to integrate numerically or any integral which cannot be even integrated numerically? Is there any limitation of numerical integration?
 A: Yes you can, but a better question is "can we integrate accurately numerically?
It is easy to write software which claims to integrate functions where the integral doesn't even exist (for example it is infinite).
If the math describes the motion of a chaotic system, any single calculated integral might not tell you much that is interesting about the real behavior of the system, and a simple integration algorithm might not even give you any clue that the system is chaotic.
There is no substitute for actually understanding what is going on! As Hamming said, "the purpose of computing is not numbers, but insight."
A: This question may be more suited on Math SE, but I will add: it depends on what you mean by limitations of numerical integration. Some problematic cases:

*

*Integrals which do not converge (e.g. $\int_0^{\infty} dx\ x$)

*Integrals which are numerically too expensive to calculate accurately (e.g. very high dimensional integrals)

*Integrals with a singularity in their integration range (e.g. $\int_0^{\infty} dx\ \frac{1}{x+1}$)

In the first case, the integral is not well defined, so it is a matter of taste whether you consider this a failure of the numerics or of the integral itself.
The second case is more of a hardware problem, and is perhaps the most common problem. This is one reason why analytic methods are crucial (e.g. in path integral computation) but analytic techniques also often fail for these integrals.
The third case is not necessarily a problem in itself, but you do have to be careful with how you sample your integration, e.g. it will depend crucially on the mesh close to the singularity. This is where asymptotic techniques become very useful.
