Rudin: chapter 6 exercise 13

**mohammadfawaz** · May 11th 2010, 02:38 AM

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

Thank you in advance.

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