Integral of $\sin (x^3)dx$ $$\int \sin (x^3)dx$$
I have tried some substitutions, but I haven't reached the goal... Can you help me?

By combining Euler's formula with the integral expression for the $\Gamma$ function, we have, for $n>1$ $$\int_0^\infty\sin(x^n)~dx~=~\bigg(\frac1n\bigg){\large!}~\sin\frac\pi{2n}$$ and $$\int_0^\infty\cos(x^n)~dx~=~\bigg(\frac1n\bigg){\large!}~\cos\frac\pi{2n}$$ Given the fact that your integral is indefinite, its expression is very similar, but it involves the incomplete $\Gamma$ function, rather than the classical one.
