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Math Help - integrate sin^2xcos^4x

  1. #1
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    Talking integrate sin^2xcos^4x

    integrate:
     sin^2xcos^4x

    at the moment, I can only think of the cos^2x = 1/2cos2x + 1/2 thing, but I think that it will make the integration look really ugly

    any other ideas for how to integrate this nicely?

    Thanks in advance
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  2. #2
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    Quote Originally Posted by differentiate View Post
    integrate:
     sin^2xcos^4x

    at the moment, I can only think of the cos^2x = 1/2cos2x + 1/2 thing, but I think that it will make the integration look really ugly

    any other ideas for how to integrate this nicely?

    Thanks in advance
    i would go with
    substitution: t = cos^3(x)

    then you need to integrate \frac{1}{3}\cdot t^{\frac{4}{3}}
    Last edited by josipive; April 2nd 2010 at 02:22 AM.
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  3. #3
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    Quote Originally Posted by josipive View Post
    i would go with
    substitution: t = cos^3(x)

    then you need to integrate \frac{1}{3}\cdot t^{\frac{4}{3}}
    Lovely idea - if only dt/dx would = 3 sin^2 x (!)

    integrate sin^2 x cos^4 x - Wolfram|Alpha
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  4. #4
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    Quote Originally Posted by differentiate View Post
    integrate:
     sin^2xcos^4x

    at the moment, I can only think of the cos^2x = 1/2cos2x + 1/2 thing, but I think that it will make the integration look really ugly

    any other ideas for how to integrate this nicely?

    Thanks in advance
    I would use the identity \sin^2 \theta + \cos^2 \theta = 1

    (1-\cos^2 x)cos^4 x = \cos^4(x) - \cos^6(x)
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