The concept of a integral curve is relatively easy to understand as path through a vector field which is tangent to the field at each point.
But why is it called an "integral" curve? It appears to have little to do with integers, or integration.
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It does somewhat have to do with integration. To find an integral curve, you have to solve a system of ODEs at every point. Let me quote from John Lee, Introduction to Smooth Manifolds, p. 207.
This is the reason for the terminology “integral curves,” because solving a system of ODEs is often referred to as “integrating” the system.
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$\begingroup$ So, when you have these differential equations which define a system, is integration always a fundamental part of their solution? I suppose it probably is. Is this the meaning of "non-integrable" if this can't be done? $\endgroup$– apkgJan 17 at 17:59
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1$\begingroup$ @apkg The standard example of a non-integrable equation is this: Can you find smooth surfaces in $\Bbb R^3$ on which we have $dz-x\,dy = 0$ at each point? (Curves are never a problem, at least locally.) $\endgroup$ Jan 18 at 3:20