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.

Since
$$|\sin x|\le1$$
and
$$\lim_{x\to\infty}\frac{\log x}{x}=0$$
then for $x$ large enough 
$$\left|\frac{\log x}{x^3}\sin x\right|\le \frac{\log x}{x^3}=_\infty o\left(\frac1{x^2}\right)$$
hence the given integral is convergent by comparison.
