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Nasty Integration problem

  1. Aug 30, 2014 #1
    1. The problem statement, all variables and given/known data

    [itex] \int sin e^{-x}+e^x cos e^{-x}\,dx[/itex]

    Find the integral above

    2. Relevant equations

    3. The attempt at a solution

    I tried substituting [itex]u=e^{-x}[/itex], but i get [itex] \int \frac{sin u}{u}+\frac{cos u}{u^2} \,du[/itex], which is non-integrable function.
    Last edited: Aug 30, 2014
  2. jcsd
  3. Aug 30, 2014 #2


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    What is the derivative of ##e^x\cos(e^{-x})##?

  4. Aug 30, 2014 #3
    Thanks. I didn't notice that
  5. Aug 30, 2014 #4


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    You are welcome.

  6. Aug 30, 2014 #5


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    A second way is to try to transform the cos into a sin function. That leads to the hope that an integration by part will work by starting with

    [itex] e^x cos (e^{-x}) = \frac{d e^x}{dx} \cos (e^{-x}) [/itex]

    and sure enough that works.
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