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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.

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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.

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