Does $\int_1^\infty \frac{\log x}{x^{3}} \sin x \,dx $ exist? I want to determine if the following indefinite integral exists:
$$\int_{1}^{\infty} \frac{\log x}{x^{3}} \sin x dx.$$
I tried to solve the integral then calculate the limit
$$ \lim_{\lambda \to \infty} ( \int_{1}^{\lambda} \frac{\log x}{x^{3}} \sin x dx ) $$
but I couldn't come to any easy way to solve the integral $\int \frac{\log x}{x^{3}} \sin x dx$ in order to calculate its limit.

Pick $x_0 \gg 1$ such that $|\log x| \leq x$ for $x > x_0$, and split $\int_1^\infty = \int_1^{x_0}+\int_{x_0}^\infty$. The first is an ordinary Riemann integral; for the second, remark that
$$
\left| \frac{\log x}{x^3} \sin x \right| \leq \frac{|\sin x|}{x^2} \leq \frac{1}{x^2},
$$
which is integrable on $(x_0,+\infty)$.
