Convergence of the improper integral $\int_1^{\infty} f(x)dx$

Question

Convergence of the improper integral $\int_1^{\infty} \cos (x^2)dx$

I tried using $x^2=t$, but I cant proof the convergence of the above integral

Answer 1

Using the substitution $t=x^2$ you get: $$\int_1^{\infty} \cos (x^2)dx=\int_1^\infty \frac{\cos x}{2\sqrt t}dt$$

This function converges by Dirichlets test. In fact $\cos x$ has a bounded antiderivative and $f(x)=\frac 1{2\sqrt t}$ is decreasing to $0$ with derivative strictly positive.