# find all the values $a,b$ for which the integral converge absolutely, converge conditionally and diverges $\int_{1}^\infty x^a\sin(x^b)$

find all the values $$a,b$$ for which the integral converge absolutely, converge conditionally and diverges: $$\int_{1}^\infty x^a\sin(x^b)$$

Hi everyone, this is a question from a PSET sheet of mine, there is a given hint which says we should divide it into two options:

• $$b\le0$$
• $$b>0$$ use $$t=x^b$$

I thought that if $$a<-1$$ than its converges absolutely regardless to the value of $$b$$ but my friend think i'm wrong. I am a bit clueless about this integral.

You are right. If $$a<-1$$ then $$|x^a\sin(x^b)|\le x^a,$$ and by the comparison test the integral converges.
If $$b\le0$$, then $$x^a\sin(x^b)\sim x^{a+b}$$ as $$x\to\infty$$, and the integral converges if and only if $$a+b<-1$$.
If $$b>0$$, the change $$x^b=t$$ transforms the integral into $$\frac1b\int_1^\infty t^{(a+1)/b-1}\sin t\,dt.$$ It converges if $$(a+1)/b-1<0$$ and diverges otherwise.