How to evaluate improper integral

Question

How to evaluate $\int_{0}^{+\infty}\frac{\sin(x^3)}{x} dx$?

It can not be represented in terms of elementary functions.

$\lim_{x \to 0} {\frac {\sin(x^3)}{x}} = 0$ and integral converges.

Help me, please!

Thanks!

Answer 1

Write the original integral as $$\int_0^\infty \frac{x^2\sin(x^3)}{x^3} \;dx$$ then let $u=x^3$ to get the integral in the form $$\frac{1}{3} \int_0^\infty \frac{\sin u}{u} \;du$$ it is well known that $$ \int_0^\infty \frac{\sin x}{x} \; dx=\frac{\pi}{2} $$ giving us the answer $$ \int_0^\infty \frac{\sin x^3}{x}\;dx=\frac{\pi}{6} $$