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Math Help - (URGENT) integration by parts

  1. #1
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    (URGENT) integration by parts

    let I = integral e^(ax) sin (bx) dx
    use integration by parts to show that

    I = e^(ax) * (a sin (bx) - b cos (bx)) / (a^2 + b^2) + C

    i have tried every possible combination of two integration by parts and have not gotten the answer provided

    can someone show me a detailed solution
    help would be greatly appreciated
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  2. #2
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    I = \int e^{ax} \sin (bx) \ dx

    Use integration by parts: u = e^{ax} and dv = \sin bx dx

    To get: I = -\frac{1}{b}e^{ax} \cos (bx) + \frac{a}{b}{\color{red}\int e^{ax}\cos (bx) dx}

    Now apply parts again to the integral in red using: u = e^{ax} and dv = \cos (bx) dx

    To get: I = -\frac{1}{b}e^{ax} \cos (bx) + \frac{a}{b}\left({\color{red}\frac{1}{b}e^{ax} \sin (bx) - \frac{a}{b} \! \! \! \! \underbrace{\int e^{ax} \sin bx dx}_{{\color{black}\text{Doesn't this look familiar?}}}}\right)

    So we really have: I = -\frac{1}{b}e^{ax} \cos (bx) + \frac{a}{b}\left(\frac{1}{b}e^{ax} \sin (bx) - \frac{a}{b}I\right)

    Now solve for I.
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  3. #3
    is up to his old tricks again! Jhevon's Avatar
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    Quote Originally Posted by razorfever View Post
    let I = integral e^(ax) sin (bx) dx
    use integration by parts to show that

    I = e^(ax) * (a sin (bx) - b cos (bx)) / (a^2 + b^2) + C

    i have tried every possible combination of two integration by parts and have not gotten the answer provided

    can someone show me a detailed solution
    help would be greatly appreciated
    this integral, and several like it, has been done many times on this forum, in many different ways.

    see here for example
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