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Math Help - Finding a recursive formula by using integration by parts

  1. #1
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    Finding a recursive formula by using integration by parts

    Does anyone know how you would go about finding the recursive formula when it is a fraction.

    E.g. Find recursive formula for 1/(x^3+1)^w (dx)
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  2. #2
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    Hello,
    Quote Originally Posted by hats_06 View Post
    Does anyone know how you would go about finding the recursive formula when it is a fraction.

    E.g. Find recursive formula for 1/(x^3+1)^w (dx)
    Integration by parts most of the time.

    Which is the case here.

    \left(\frac{1}{(x^3+1)^w}\right)'=\left((x^3+1)^{-w}\right)'=\frac{-3wx^2}{(x^3+1)^{w+1}}=du

    dv=1 \implies v=x

    I_w=\int \frac{1}{(x^3+1)^w} ~dx=\frac{x}{(x^3+1)^w}+3w \int \frac{x \cdot x^2}{(x^3+1)^{w+1}} ~dx

    =\frac{x}{(x^3+1)^w}+3w \int \frac{x^3}{(x^3+1)^{w+1}} ~dx

    Write x^3=(x^3+1)-1 and remember that I_{w+1}=\int \frac{1}{(x^3+1)^{w+1}} ~dx after that, it's just algebra and you're done
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  3. #3
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    can you do it writing it like this?

    (x^3+1)^-w (dx)

    sorry man i havent done recursions for a while...does that help?

    EDIT: beaten to it...i was nowhere near as much help.
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  4. #4
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    Hey moo thanks for your help! I had gotten it right up until this point:

    Write and remember that after that, it's just algebra and you're done

    I dont still quite understand why
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  5. #5
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    Oh wait i understand!! Thanks so much once again!
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  6. #6
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    Heey its me again...I got slightly confused again when i evaluated it...but i ended up with this:

    Iw = 3wIw - 3wIw+1 + (x)/(x^3+1)^w


    Does that sound right?
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  7. #7
    Moo
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    Quote Originally Posted by hats_06 View Post
    Heey its me again...I got slightly confused again when i evaluated it...but i ended up with this:

    Iw = 3wIw - 3wIw+1 + (x)/(x^3+1)^w


    Does that sound right?
    Perfect

    Now, write I_{w+1} in terms of I_w (that's not much, it's just for aesthetics ^^)
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  8. #8
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    So just:

    Iw+1 = Iw - (1/3w)Iw + (x)/3w(x^3+1)^w

    Is that it?
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  9. #9
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    Yes it is

    or more precisely (1-1/3w)I_w
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  10. #10
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    Thanks so much my genius friend!
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