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Math Help - Integrate sin^2(x)*cos^2(x)

  1. #1
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    Integrate sin^2(x)*cos^2(x)

    Hi can someone please help me integrate sin^2(x)*cos^2(x). It has something to do with trig identities (obviously). I tried using the double angle identities but just made more of a mess. Even if someone could show me what how to manipulate the trig stuff into something easier to integrate and then I will have a crack at the integration myself.

    Thank you in advance
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  2. #2
    Super Member Random Variable's Avatar
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     \cos^{2}x \sin^{2}x = \Big(\frac {1}{2} + \frac {\cos2x}{2}\Big)\Big(\frac {1}{2} - \frac {\cos2x}{2} \Big) = \frac {1}{4} - \frac {1}{4} \cos^{2} 2x = \frac {1}{4} - \Big(\frac {1}{8} - \frac{1}{8} \cos 4x \Big)  = \frac {1}{8} -  \frac{1}{8} \cos 4x
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  3. #3
    MHF Contributor alexmahone's Avatar
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    sin^2 (x)cos^2(x)=(sin x cos x)^2

    = (\frac{1}{2}sin (2x))^2

    = \frac{1}{4}sin^2(2x)

    = \frac{1}{4}\frac{1-cos (4x)}{2}

    = \frac{1}{8}(1-cos (4x))

    -------------------------------------------------------------------

    \int sin^2 x cos^2 x=\int \frac{1}{8}(1-cos (4x))

    = \frac {1}{8}(x-\frac{sin (4 x)}{4})

    = \frac{1}{8}\frac{4x-sin (4x)}{4}

    = \frac{1}{32} (4x-sin (4x))

    You can verify the answer using Wolfram Mathematica Online Integrator
    Last edited by alexmahone; July 5th 2009 at 09:51 PM.
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  4. #4
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    Quote Originally Posted by alexmahone View Post
    sin^2 (x)cos^2(x)=(sin x cos x)^2

    = (\frac{1}{2}sin (2x))^2

    = \frac{1}{4}sin^2(2x)

    = \frac{1}{4}\frac{1-cos (4x)}{2}

    = \frac{1}{8}(1-cos (4x))

    -------------------------------------------------------------------

    \int sin^2 x cos^2 x=\int \frac{1}{8}(1-cos (4x))

    = \frac {1}{8}(x-\frac{sin (4 x)}{4})

    = \frac{1}{8}\frac{4x-sin (4x)}{4}

    = \frac{1}{32} (4x-sin (4x))

    You can verify the answer using Wolfram Mathematica Online Integrator
    Thank you very much for the time you put into that reply. It really makes sense now.
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