I am posting here a problem from my homework. I am having trouble with a number of problems, but I think guidance on this one should help me grasp the general concept and complete some of the others.
Here is the problem:
Find, with proof, all p∈R for which the Lebesgue integral $\int_0^{\infty} \! x^p\mathrm{sin}(x^2) \, \mathrm{d}x$ exists.
Now, I understand that for p>0, $x^p$ diverges so the integral does not exist. Then for p=0, $\int_0^{\infty} \! |\mathrm{sin}(x^2)| \, \mathrm{d}x$ diverges so the Lebesgue integral does not exist.
My trouble lies in p<0. I understand that close to 0, the sin term approaches 0, but the $x^p$ term approaches $\infty$. In some range of values, these effects will cancel out, making this integral exist, but I can't figure out how to determine this range.
Thank you in advance for the help!
As x approaches 0 have a look at the power series of $$\sin(x^2)$$ It starts with $$x^2+x^6/6!$$ Thus you can consider the behaviour of $$x^{p+2}(1+O(x^4))$$ for x close to 0.