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Math Help - [SOLVED] Integral of cos squared x

  1. #1
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    [SOLVED] Integral of cos squared x

    \int sin^{2}xcosx dx

    so using
     sin^{2}x = 1 - cos^{2}x
    \int sin^{2}xcosx dx = \int cosx - cos^{3}x dx

    which is all fine and dandy (i think) but what's the integral of
    cos^{3}x ?

    is there a better way of doing it?

    Also what would be
    \int sin^{5}x dx or \int cos^{5}x dx and \int cos^{7}x dx

    i guess im looking for a general rule of
    \int cos^{n}x dx and \int sin^{n}x dx

    thanks in advance




    Last edited by UbikPkd; October 13th 2007 at 01:02 AM.
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  2. #2
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    Quote Originally Posted by UbikPkd View Post
    \int sin^{2}xcosx dx

    so using
     sin^{2}x = 1 - cos^{2}x
    \int sin^{2}xcosx dx = \int cosx - cos^{3}x dx

    which is all fine and dandy (i think) but what's the integral of
    cos^{3}x ?

    thanks in advance


    INT.[sin^2(X)cosX]dX
    = INT.[sin^2(X)]cosX dX

    if u were sinX, then du = cosX dX, so

    = (1/3)sin^3(X) +C
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  3. #3
    Senior Member DivideBy0's Avatar
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    I personally would use u-substitution. Notice that \cos{x} is the derivative of \sin{x}, so it works nicely.

    Let u=\sin{x}, then \,du=\cos{x}\,dx

    So that turns the integral into

    \int u^2 \,du=\frac{u^3}{3}+C

    Now substituting \sin{x} back into u:

    =\frac{(\sin{x})^3}{3}+C

    (Didn't see u post ticbol) :P
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  4. #4
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    thats great, thanks a lot!
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  5. #5
    Senior Member DivideBy0's Avatar
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    Unfortunately there is no easy generalisation for nth powers, you have to use reduction formulas. Here are some of them. You could try to derive them using integration by parts, if you've learnt that.
    Attached Thumbnails Attached Thumbnails [SOLVED] Integral of cos squared x-reduction-formulae.jpg  
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  6. #6
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    i had a feeling it wasn't gonna be simple....thanks anyway!
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