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Integration Question (In a loop)

  1. Feb 25, 2007 #1
    1. The problem statement, all variables and given/known data
    e^x * sin(x)

    2. Relevant equations
    uv - integral vdu

    3. The attempt at a solution
    FIRST, I set u = e^x, du = e^x, dv = sin(x)dx, v = -cos(x)

    Using, uv - integral vdu

    e^x * -cos(x) - [ e^x * -cos(x)dx

    -e^x * cos(x) + [ e^x * cos(x)

    Now I'm stuck... if I keep integrating... I'll keep going in a loop of sin/cos(x) and e^x.

    Am I going about this in the wrong way?
  2. jcsd
  3. Feb 25, 2007 #2


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    Staff Emeritus
    Science Advisor

    You don't actually state that you are trying to integrate exsin(x) but I will assume that!

    One thing that is confusing you is that you are not writing out full equations!

    After your first integration by parts, you get
    [tex]\int e^x sin(x)dx= -e^x cos(x)+\int e^x cos(x) dx[/tex]

    Now let u= ex, dv= cos(x), so that du= exdx and v= sin(x) and you get
    [tex]\int e^x sin(x)dx= -e^x cos(x)+ e^x sin(x)- \int e^x sin(x)dx[/tex]

    Is that the "loop" you mean? Now solve that equation for
    [tex]\int e^x sin(x)dx[/tex]!
  4. Feb 26, 2007 #3


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    Homework Helper

    [tex] \sin x= \mathfrak{Im} \left(e^{ix}\right) [/tex]

    and the integral becomes a simple exponential.
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