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Consider the asymptotic expansion of the following integral as $x\rightarrow\infty$:

$$ \int_{0}^{\pi} cos(xt)sin^{2}(t^2)dt. $$

I'm trying to compute as many terms as possible using integration by parts. I.e. performing integration by parts on this until it fails. The boundary terms then form an asymptotic expansion for the integral. However, when I do it myself, I'm finding that I can compute infinite terms--but this must be wrong. Can anyone show me how to do this?

Edit 1: This is problem 6.19(b) in the book by Bender and Orszag if anyone was curious.

Edit 2: I'm getting an answer that looks like:

$$ \frac{sin(x\pi)sin(\pi^{2})}{x}+\frac{cos(x\pi)sin(\pi^{2})}{x^{2}}+\frac{sin(x\pi)sin(\pi^{2})}{x^{3}}+\cdots $$

Is this reasonable? I'm not sure if I'm going wrong somewhere...

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Indeed it would be weird if you could compute infinite terms, but on the assumption that you meant infinitely many terms: Why do you think you shouldn't be able to do that? –  joriki Oct 28 '12 at 3:45
    
@joriki: Haha, good catch, that is indeed what I meant. The question asks me to actually compute the terms, not just say how many I can compute. I have a feeling it's a finite number, but I could be wrong. –  Alex Oct 28 '12 at 3:47

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