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Math Help - Integration problem

  1. #1
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    Integration problem

    i have to integrate e^(2x) * cos(3x) dx by parts.

    so far, I have u = e^(2x), u' = 2e^(2x), v' = cos(3x), and v = 1/3 sin (3x)

    This means that the integral of e^(2x) * cos(3x) dx is:

    e^(2x) * 1/3 sin (3x) - the integral of 2e^(2x) * 1/3 sin(x)

    I'm not exactly sure where to go from here, so any help would be appreciated!
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  2. #2
    is up to his old tricks again! Jhevon's Avatar
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    Quote Originally Posted by clockingly View Post
    i have to integrate e^(2x) * cos(3x) dx by parts.

    so far, I have u = e^(2x), u' = 2e^(2x), v' = cos(3x), and v = 1/3 sin (3x)

    This means that the integral of e^(2x) * cos(3x) dx is:

    e^(2x) * 1/3 sin (3x) - the integral of 2e^(2x) * 1/3 sin(x)

    I'm not exactly sure where to go from here, so any help would be appreciated!
    Personally, I would have chosen e^(2x) as the function to integrate, but let's do it your way.

    int{e^(2x)cos(3x)}dx
    = (1/3)sin(3x)e^(2x) - int{2(1/3)sin(3x)e^(2x)}dx
    = (1/3)sin(3x)e^(2x) - (2/3)int{sin(3x)e^(2x)}dx .........do this last integral by parts again
    = (1/3)sin(3x)e^(2x) - (2/3)[(-1/3)cos(3x)e^(2x) - int{2(-1/3)cos(3x)e^(2x)}dx]
    = (1/3)sin(3x)e^(2x) - (2/3)[(-1/3)cos(3x)e^(2x) + (2/3)int{cos(3x)e^(2x)}dx]
    = (1/3)sin(3x)e^(2x) - [(-2/9)cos(3x)e^(2x) + (4/9)int{cos(3x)e^(2x)}dx]
    = (1/3)sin(3x)e^(2x) + (2/9)cos(3x)e^(2x) - (4/9)int{cos(3x)e^(2x)}dx

    So now we have:
    int{e^(2x)cos(3x)}dx = (1/3)sin(3x)e^(2x) + (2/9)cos(3x)e^(2x) - (4/9)int{cos(3x)e^(2x)}dx

    So we solve this equation for int{e^(2x)cos(3x)}dx

    => int{e^(2x)cos(3x)}dx + (4/9)int{cos(3x)e^(2x)}dx = (1/3)sin(3x)e^(2x) + (2/9)cos(3x)e^(2x)
    => (13/9)int{e^(2x)cos(3x)}dx = (1/3)sin(3x)e^(2x) + (2/9)cos(3x)e^(2x)
    => int{e^(2x)cos(3x)}dx = (9/13)[(1/3)sin(3x)e^(2x) + (2/9)cos(3x)e^(2x)]
    => int{e^(2x)cos(3x)}dx = (3/13)sin(3x)e^(2x) + (2/13)cos(3x)e^(2x) + C


    Also, see the last note of this post: http://www.mathhelpforum.com/math-he...l-problem.html
    Last edited by Jhevon; March 3rd 2007 at 06:47 PM.
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