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Math Help - trigonometric integral

  1. #1
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    trigonometric integral

    integral of cos(x)sin(2x)
    the answer is (-1/6)cos(3x)-.5cos(x)
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  2. #2
    MHF Contributor kalagota's Avatar
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    Quote Originally Posted by xslim12 View Post
    integral of cos(x)sin(2x)
    the answer is (-1/6)cos(3x)-.5cos(x)
    \int cos(x)sin(2x)

    sin (2x) = 2sinxcosx
    so,
    \int cos(x)sin(2x) = \int 2cos^2(x)sinx

    let u = cos x
    so, du =-sinx dx

    from here, u can continue it..
    should the answer be like that?
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  3. #3
    Forum Admin topsquark's Avatar
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    Quote Originally Posted by xslim12 View Post
    integral of cos(x)sin(2x)
    the answer is (-1/6)cos(3x)-.5cos(x)
    sin(a + b) = sin(a)cos(b) + sin(b)cos(a)
    sin(a - b) = sin(a)cos(b) - sin(b)cos(a)

    Thus
    sin(a + b) + sin(a - b) = 2sin(a)cos(b)

    So
    sin(a)cos(b) = \frac{sin(a + b) + sin(a - b)}{2}

    Your integral, then is
    <br />
\int sin(2x)cos(x)~dx = \frac{1}{2} \int (sin(3x) + sin(x) )~dx

    -Dan
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  4. #4
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    Quote Originally Posted by topsquark View Post
    sin(a + b) = sin(a)cos(b) + sin(b)cos(a)
    sin(a - b) = sin(a)cos(b) - sin(b)cos(a)

    Thus
    sin(a + b) + sin(a - b) = 2sin(a)cos(b)

    So
    sin(a)cos(b) = \frac{sin(a + b) + sin(a - b)}{2}

    Your integral, then is
    <br />
\int sin(2x)cos(x)~dx = \frac{1}{2} \int (sin(3x) + sin(x) )~dx

    -Dan
    thanks man. blahh i need to start realizing these things.
    Kalgota, i havn't tried your method yet, but i found this much simpler. thanks for the help both.
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  5. #5
    Forum Admin topsquark's Avatar
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    Quote Originally Posted by xslim12 View Post
    thanks man. blahh i need to start realizing these things.
    Kalgota, i havn't tried your method yet, but i found this much simpler. thanks for the help both.
    Yeah, but what you don't know is how many problems I ran into that I couldn't integrate before I finally started remembering this trick!

    You should try kalagota's method also. It's always good to practice trig integrals (since they are, in my opinion, the nastiest kind.)

    -Dan
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