How do you know when an expression is not solvable?

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Determining whether an expression is solvable often requires specific mathematical theories, such as differential Galois theory, which assesses the solvability of integrals and differential equations. For instance, the integral of e^{-x^2} cannot be expressed in terms of elementary functions, indicating its unsolvability. It is challenging to ascertain the solvability of an integral without prior proof or established knowledge. Numerical methods may be necessary when analytical solutions are not feasible. Understanding these concepts is crucial for navigating complex mathematical problems effectively.
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My question pretty much is the title of this post, but let me explain it a bit more. When in physics or mathematics, or some other discipline involving math, you run into an integral, a differential equation or some other expression, how do you know if it's solvable or not? I mean, how do I know when to tend to numerical computations because an analytical result cannot be obtained? How do I know that my incapability to solve a certain equation is not just due to my lack of knowledge, but because the equation is, in fact, now solvable?

For example, how would I know that the integral:

<br /> \int e^{-x^2} dx<br />

is not solvable?

Thanks in advance
 
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I don't think there is anyway of knowing whether an integral is possible or not unless it's been proven one way or another for that specific integral.
 
It's usually very difficult. You know how Galois theory helps us determine the solvability of polynomial equations? Well, there is a field of math known as differential Galois theory that can be used to prove that the integral you quoted (for instance) is not solvable, in the sense that it cannot be expressed in terms of elementary functions. Although personally I'm ignorant of its ways!

You can also read the following pdf (which AFAIK was posted on these forums):
http://www.claymath.org/programs/outreach/academy/LectureNotes05/Conrad.pdf
 
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