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Math Help - Rudin: chapter 6 exercise 13

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    Member mohammadfawaz's Avatar
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    Rudin: chapter 6 exercise 13

    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.
    Last edited by mohammadfawaz; May 11th 2010 at 04:32 AM.
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