Show $\int \frac{\sin(x^p)}{x} dx = \frac{\operatorname{Si}(x)}{p} $

Question

I was messing around with the Fresnel integral and the Sine integral and found that $\int_{0}^{\infty} \frac{\sin(x^2)}{x}dx=\frac{\pi}{4}$ but I dont see how to extend to irrational powers.

Answer 1

Using the substitution $y=x^p$, for $p>0$, $$\int_0^\infty\sin(x^p)\frac{dx}{x} =\int_0^\infty\sin(y)\frac{dy}{py}=\frac\pi{2p}.$$