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Integration by parts computation

  1. Oct 26, 2012 #1
    1. The problem statement, all variables and given/known data
    Consider the following integral:

    I=[itex]\int^{\pi/4}_{0}[/itex]cos(xt[itex]^{2}[/itex])tan[itex]^{2}[/itex](t)dt

    I'm trying to compute as many terms as possible of its asymptotic expansion as x[itex]\rightarrow\infty[/itex].


    2. Relevant equations

    x

    3. The attempt at a solution
    Let u=cos(xt[itex]^{2}[/itex]). And dv=tan[itex]^{2}[/itex](t)dt.
    Then du=-2xtcos(xt[itex]^{2}[/itex])dt and v=[itex]\int[/itex]tan[itex]^{2}[/itex](t)dt=tan(t)-t+C.

    Integration by parts yields:

    I=cos(xt[itex]^{2}[/itex])tan(t)-tcos(xt[itex]^{2}[/itex])+[itex]\int[/itex][2xtcos(xt[itex]^{2}[/itex])tan(t)-2xt[itex]^{2}[/itex]cos(xt[itex]^{2}[/itex])]dt,
    where all terms are evaluated from 0 to pi/4, obviously.

    This feels wrong to me. Can anyone give me some help?
     
  2. jcsd
  3. Oct 27, 2012 #2

    tiny-tim

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    hi the_kid! :smile:

    (try using the X2 button just above the Reply box :wink:)
    erm :redface:

    du=-2xtsin(xt2)dt :wink:
     
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