All integrals exist, of course. But the bulk of them can only be done numerically. The answer to your question is lots and lots of practice. You learn to recognize specific forms over time, those that can be done and those that can't. Certain fields of study usually have typical types of integrals and you can specialize from those. For example, in Mechanics, common undoable integrals are those of the form of elliptic integrals. Similar comments also apply to Differential Equations, which can be seen as a kind of integration.
Remember those integral tables you thought I was being facetious about? If you learn how to do them you will have a good grasp of which integrals can be done.
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