Rudin: chapter 6 exercise 13

Feb 2010
Lebanon - Beirut
I'm stuck on the following problem from Rudin:
we're given: \(\displaystyle f(x)=\int_x^{x+1} sin(t^2)dt\).
The first question was to prove that \(\displaystyle |f(x)|<\frac{1}{x}\) if \(\displaystyle x>0\). This was easy to prove if we let \(\displaystyle u=t^2\) and then integrate by parts.
The second question is to show that \(\displaystyle 2xf(x)=cos(x^2)-cos[(x+1)^2]+r(x)\) where \(\displaystyle |r(x)|<\frac{c}{x}\) where c is a constant. Well, that was also easy to prove and we get c = 2.
Now, the third parts asks for the upper and lower limits of \(\displaystyle xf(x)\) as \(\displaystyle x\rightarrow\infty\). How can we do this?
Finally, we're asked whether \(\displaystyle \int_0^{\infty}sin(t^2)dt\) converges or not. How can we know?

Thank you in advance.
Last edited: