Hint: First prove that $f = ( x \mapsto x \ln(x) )$ is an increasing function on $[1,\infty)$. Next let $y$ be the increasing sequence such that $f(x) = n π$ for any $n \in \mathbb{N}^+$, and prove that $y_{n+1}-y_n \to 0$ as $n \to \infty$. Then since $f$ is increasing show that the sequence $A = ( \int_{y_n}^{y_{n+1}} \sin(f(x))\ dx )_{n\in\mathbb{N}^+}$ is alternating with absolute value going to zero. Thus the integral $\int_1^\infty \sin(f(x))\ dx$ converges, as it is bounded between adjacent partial sums of $A$.
http://www.wolframalpha.com/input/?i=Integrate[Sin[x^2],{x,0,Infinity}]
for an exact value! Edit: Crostul removed his comment; his claim was that the integral does not converge because the integrand does not converge to zero. I'll leave this here in case other people make the same mistake. $\endgroup$