Investigate convergence with cos/sin

Question

Well I want to investigate the convergence of the following integrals(in the linked picture): $$\int_{1}^{\infty}\cos(x^t)dx\quad,\quad t\in \mathbb{R}$$ $$\int_{1}^{\infty}\sin(x^t)dx\quad,\quad t\in \mathbb{R}$$

for example , $$\int_{1}^{\infty}\cos\sqrt[4]{x^3}\,dx$$

Can someone throw a bone to how do I deal with this kind of problems?

Answer 1

Enforcing the substitution $x\to x^{1/t}$, we have

$$\int_1^\infty \cos(x^t)\,dx=\frac1t\int_1^\infty \frac{\cos(x)}{x^{1-1/t}}\,dx$$

which converges provided $t>1$ using the Dirichlet Test.