Noncritical Level Lines of Energy in Arnold' book about ODEs

Question

I am trying to understand the follwoing from Arnold's book about ODEs:

In the second to last line it says that "the integral converges, since $U'(a),U'(b)\neq 0$". Why do we need that for the convergence? But doesn't he actually mean just the existence of the intergral

Many Thanks in advance!

Answer 1

Since $U(a)=U(b)=E$ you get that $$ E-U(\xi)=U'(a)(\xi-a)+\frac{U''(a)}{2}(\xi-a)^2+\dots, $$ If you assume that $U'(a)=0$ then the corresponding improper integral diverges.