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Math Help - integration of binomial expansion

  1. #1
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    integration of binomial expansion

    how would you integrate the general binomial expansion??

    thanks for and help
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  2. #2
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    \int (a\times x+b)^n \, dx=\frac{(a x+b)^{n+1}}{a(n+1)}

    If F (x) = \int f(x)dx then:
    F(ax+b)=F(x)/a
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  3. #3
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    thanks

    thanks but i was more asking how you would integrate the right hand side of the equation? sorry
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  4. #4
    MHF Contributor Mathstud28's Avatar
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    Quote Originally Posted by hmmmm View Post
    thanks but i was more asking how you would integrate the right hand side of the equation? sorry
    Since the sum is finite you may interchange the summation and the integral so \int\sum_{k=0}^{n}{n\choose{k}}x^{n-k}y^k=\sum_{k=0}^{n}\int\left\{{n\choose{k}}x^{n-k}y^k\right\}
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  5. #5
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    thanks

    so how do we integrate this?
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  6. #6
    Member Greengoblin's Avatar
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    \sum_{k=0}^{n} {n\choose{k}}x^{n-k}y^k

    = x^n+nx^{n-1}y+\frac{n(n-1)}{2!}x^{n-2}y^2+\cdots +\frac{n(n-1)\cdots(n-r+1)}{r!}x{n-r}y^r+\cdots+y^n

    So taking:

     \int\sum_{k=0}^{n}{n\choose{k}}x^{n-k}y^kdx

    Is just a case of using the addition and power rules for integrals on this series.
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