Absolute and conditional convergence of an integral $\int_1^\infty x^p\cos(\ln x)\,dx$

Question

$$\int_1^\infty x^p\cos(\ln x)\,dx$$

Need to find values of $p\in \Bbb R$ for which the integral converges 1) absolutely and 2) conditionally.

I've used partial integration and get that it converges for $p<0$, but not sure about that.

Thanks.

Answer 1

if the integrand is positive $|x^p\cos(\ln(x))|\le x^p$ which converges iff $p<-1$ (we know it by the series test with $\displaystyle\sum_{k=1}^{\infty}\frac 1 {n^p}$). Ao what we got is inetgral converges absolutely iff $p<-1$

if $-1\le p< 0$ you can use dirichle test: $\lim_{x\to\infty}\frac 1 {x^{-p}}=0$ and $\int_a^bcos(ln(x))dx\le\int\cos(x)dx<\infty$. The derivativeof $x^p$ is continious and cos is obviously continious so we got all th term fullfiled.

As for $p>0$, you can show directly (e.g by partial integration) the integral diverges (take the limit in $\infty$).

Conclusion: we got the integral absouetly converges is $p<-1$, conditionally converges is $-1\le p<0$ and diverges otherwise.