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    • Thread Starter

    Using integration by parts, prove that

    integral(e^(ax)sinxdx)= ((e^(ax))/(1+a^2))(asinx-cosx))

    ive been on this for some time now, please help.

    Have you tried applying integration by parts twice?
    Notice that if you integrate/differentiate the 'sin' part twice you'll end up with sin again, and similarly you'll get e^(ax) again after differentiating/integrating e^(ax) twice .There'll be some constants in this, this doesn't matter, what you're meant to see is that by the integration by parts formula you'll eventually end up with the same integral you started with.
    That's a bit wordy, so if you can't guess what to do here's some hints:

    Let  I = \int e^{ax} \sin x \ dx .

    Integrating by parts where you differentiate e^(ax) and integrate sin will give:

     I = -e^{ax} \cos x + a \int e^{ax} \cos x \ dx

    Make sure you can get this (note that it doesn't matter if you choose to differentiate sin and integrate e^(ax), but you'll probably get a different expression to me).

    Now apply integration by parts again to that second integral.
    You should get
     I = -e^{ax} \cos x + ae^{ax} \sin x - a^2 I

    The answer follows from that
    • Thread Starter

    awww i did a similar thing but i didn't take a out of the second integral. This is now working. thank you for your help
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Updated: February 4, 2010

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