# Determine for which $t\in \mathbb{R}$ the integral $\int_0^\infty \sin(x^t)\ dx$ converges!

Obviously for $$t=0$$ the integral diverges.

For $$t>1$$ I have tried to subsitute $$u=x^t$$ and then split the integral up into segments of pi length such that I can write it as an alternating sum. Then (I think) I managed to prove that I can use Leibniz test since the alternating integrals get ever so smaller as $$x \to \infty$$.

Also for $$x\in(0,1]$$ I can bound the sine from below by inscribing ever increasing triangles thus proving that the sequence doesn't go to 0, which implies that the series is divergent.

Now for $$t<0$$ I am completely clueless, partly because the graph is vastly different in that case. Substituting $$1/x = u$$ gets ugly very quickly, so I didn't follow through.

I'd highly appreciate any ideas and solutions, particularly if they are more elegant than my tries so far have been :)

Thanks in advance!

• You might want to look at this and this. – Mattos Nov 3 '18 at 15:24
• Unfortunately I must not use the Dirichlet for integrals for I have not learned it yet. But thank you for the great resources - odd that I didn't find them myself. – whiterock Nov 3 '18 at 15:28
• Ignore the second link, the first one gives you what you want. There is also a good way to search the site for specific math text, I believe it was written by a user of the site who got annoyed with the in built MSE search function. It's called approach0. – Mattos Nov 3 '18 at 15:29
• As I said, alas I cannot use the Dirichlet test. – whiterock Nov 3 '18 at 15:33
• The first link doesn't use the Dirichlet test? – Mattos Nov 3 '18 at 15:37