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

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    Integrate (sin(x))^2(cos(x))^2

    As the title says, I need help with Integrate (sin(x))^2(cos(x))^2

    I get thet I need to use the half- angle formula, but do I use it on both sin(x)^2 and cos(x)^2? And where do I go from there. Much appreciated!
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    Quote Originally Posted by birdz View Post
    As the title says, I need help with Integrate (sin(x))^2(cos(x))^2

    I get thet I need to use the half- angle formula, but do I use it on both sin(x)^2 and cos(x)^2? And where do I go from there. Much appreciated!
    If you use the half-angle/power reduction formula on both factors from the start, you can then expand and use the formula again.

    Alternatively, you could first use the Pythagorean identity, \sin^2\theta+\cos^2\theta=1 to get everything into terms of either sine or cosine. Then you could apply the half-angle formula twice. I don't think this would be any easier, but it is certainly valid.
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    \left[\frac{1 - \cos(2x)}{2}\right] \cdot \left[\frac{1 + \cos(2x)}{2}\right] =

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

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

    \frac{1}{4}\left[\frac{1 - \cos(4x)}{2}\right] =

    \frac{1}{8}\left[1 - \cos(4x)\right] <br />

    integrate the last expression
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    Thank you for your very quick replies! I get it now
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    It would have been better to directly note that \sin(x)\cos(x)=\frac 12 \sin(2x), and then use the half angle formula
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