For what $a$ does this $\int_0^{+\infty}\frac{\sin(x)\cos(1/x)}{x^a}$ converge?

Question

For what $a$ does this $\int_0^{+\infty}\frac{\sin(x)\cos(1/x)}{x^a}$ converge?

I’m preparing for a test and this exercise has me wondering: I though about breaking it up to $(0,1)$ and $(1,+\infty)$ interval but I don’t know how to do the $(0,1)$ part, the $(1,+\infty)$ should be doable by Abel-Dirichlet criteria.

Robert Z · Accepted Answer · 2018-04-22T13:32:48.570

0

Hint. Consider two improper integrals: $$\int_1^{+\infty}\sin(x)\cdot \frac{\cos(1/x)}{x^a}\,dx$$ and $$\int_0^1\frac{\sin(x)\cos(1/x)}{x^a}\,dx=\int_1^{+\infty}\cos(t)\cdot\frac{\sin(1/t)}{t^{2-a}}dt$$ where $t=1/x$. Now use Dirichlet's test for convergence of improper integrals

edited Apr 22 '18 at 13:32

answered Apr 22 '18 at 13:26

Robert Z

145,942

Thanks for the answer, I would like to know in the second integral why there is t^(2-a) in the deniminator? – innerz09 Apr 22 '18 at 13:34
After letting $t=1/x$ then $dx=-dt/t^2 $. – Robert Z Apr 22 '18 at 13:36
I literally always forget about that when substituting, thanks a lot! Have a nice day – innerz09 Apr 22 '18 at 13:37

For what $a$ does this $\int_0^{+\infty}\frac{\sin(x)\cos(1/x)}{x^a}$ converge?

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