Proof the convergenсe of the integral: $ \int\limits_{1}^{\infty} \sin (x \ln x) d x.$ Tried to change variables, no result. May you suggest something? And may you also suggest some books with a focus on examples? Feeling trouble with this type of integrals.
2026-04-13 04:02:29.1776052949
Does the improper integral converges?
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Hint: Let $f(x) = x\ln x.$ Then $f'(x) \ge 1,$ so $f$ is a nice 1-1 map from $[1,\infty)$ t0 $[0,\infty).$ Make the change of variables $x= f^{-1}(y).$ You get
$$\int_0^\infty (\sin y)(f^{-1})'(y)\,dy.$$
You are set up for Dirichlet's test.