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Math Help - Integration - Reduction Method: PLEASE HELP!!!

  1. #1
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    Exclamation Integration - Reduction Method: PLEASE HELP!!!

    I've attempted both questions but I cant seem to get the answers. Could someone please show me where I got it wrong.

    P.S. For the question that's typed up, I'm trying to find out the last part of the question i.e. "evaluate ..."
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  2. #2
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    Quote Originally Posted by xwrathbringerx View Post
    I've attempted both questions but I cant seem to get the answers. Could someone please show me where I got it wrong.

    P.S. For the question that's typed up, I'm trying to find out the last part of the question i.e. "evaluate ..."
    I_{n+1} = \int \frac{\cos^{2(n+1)} x}{\sin x} ~dx = \cos^2 x \int \frac{\cos^{2n} x}{\sin x} ~dx =  I_n \cos^2 x = I_n - I_n \sin^2 x

    I_n \sin^2 x = \int \frac{\sin^2 x\cos^{2n} x}{\sin x} ~dx = \int \sin x \cos^{2n} x ~dx = - \frac{\cos^{2n+1} x}{2n + 1}

    hence,

    I_{n+1} = I_n - I_n \sin^2 x = I_n + \frac{\cos^{2n+1} x}{2n + 1}

    and the result follow
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  3. #3
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    Unhappy

    I cant seem to be able to get the next part at all...
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