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Math Help - integration

  1. #1
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    integration

    How do you integrate (e^x)*cos x dx?
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  2. #2
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    Quote Originally Posted by gwiggy View Post

    How do you integrate (e^x)*cos x dx?
    a boring and over-used way is to use by parts twice! a much neater way is to use this fact that \cos x = \frac{e^{ix} + e^{-ix}}{2} and you'll get the answer very quickly, which is: \frac{e^x(\cos x + \sin x)}{2} + C.
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  3. #3
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    Edit: Excuse me for being boring. xp I'll have to remember to use that.

    Integration by parts should work fine.

    \int udv = uv - \int vdu

    u = e^x ... du = e^x
    dv = cosxdx ... v = sinx

    \int (e^x)cosxdx = (e^x)sinx - \int (e^x)sinxdx

    Doesn't look like it helps, does it? But, if we use integration by parts again for (e^x)sinxdx

    \int (e^x)cosxdx = (e^x)sinx - (-(e^x)cosx - \int -(e^x)cosxdx)

    \int (e^x)cosxdx = (e^x)sinx + (e^x)cosx - \int (e^x)cosxdx

    Now some simple algebra, add \int (e^x)cosxdx to both sides

    2\int (e^x)cosxdx = (e^x)sinx + (e^x)cosx

    I trust you can take it from there.
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