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Math Help - Indefinite Integration Problem

  1. #1
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    Indefinite Integration Problem

    I'm not sure how to approach this...

    \int e^{2x}sin(3x)

    I don't know where to start. This is on an assignment largely based on integration by parts, but I can't figure out what to do. If I do integration by parts, I still have to integrate

    \int e^{2x}cos(3x)

    so I'm not getting any closer to an answer.

    Using online solving tools either gets me no answer, or something like this;

    \dfrac{e^{2x}(2sin(3x) - 3cos(3x))}{13}

    But it isn't much of a hint. Can someone point me in the right direction?
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  2. #2
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    Quote Originally Posted by Sucker Punch View Post
    I'm not sure how to approach this...

    \int e^{2x}sin(3x)

    I don't know where to start. This is on an assignment largely based on integration by parts, but I can't figure out what to do. If I do integration by parts, I still have to integrate

    \int e^{2x}cos(3x)

    so I'm not getting any closer to an answer.

    Using online solving tools either gets me no answer, or something like this;

    \dfrac{e^{2x}(2sin(3x) - 3cos(3x))}{13}

    But it isn't much of a hint. Can someone point me in the right direction?

    Do integration by parts TWICE, choosing v'=e^{2x} in both cases, and then pass a summand from the RHS to the LHS...and voila!

    Tonio
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  3. #3
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    Quote Originally Posted by tonio View Post
    Do integration by parts TWICE, choosing v'=e^{2x} in both cases, and then pass a summand from the RHS to the LHS...and voila!

    Tonio
    Ah, of course. Thank you very much, I got it.
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  4. #4
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    Then there is the formula:

    \displaystyle \int e^{ax}\sin(bx) dx = e^{ax}\left\{\frac{a\sin{bx}-b\cos{bx}}{a^2+b^2}\right\}+k

    Similarly:

    \displaystyle \int e^{ax}\cos(bx) dx = e^{ax}\left\{\frac{a\cos{bx}+b\sin{bx}}{a^2+b^2}\r  ight\}+k

    They can be derived using by-parts twice -- same as the problem.
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