Show the improper integral $\int_0^\infty \frac{\sin(x^3-ax)}{x}dx$ converges. This is an exercise for the Dirichlet test for the convergence of improper integrals. This exercise is from the book called “A Course of Modern Analysis”. It does not specify what $a$, but I am thinking $a$ is a nonnegative number. I am not sure if this assumption is legit. Clearly, $\frac{1}{x}$ converges to 0 as $x \to \infty$. I am struggle to show $|\int_0^L \sin(x^3-ax)dx|$ for all $L$ is bounded. I tried to use u-sub but it does not make the problem easier. Any hint will be appreciated.
hint
$$\sin(x^3-ax)=\sin(x^3)\cos(ax)-\cos(x^3)\sin(ax)$$
Hint. Note that $$\left|\frac{\sin (x^3-ax)}{x}\right|\le \frac1x$$ for all $x&gt;0.$