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This integral solveable [itex]\int^{3.5}_{0.5}(sin(x*π)+1)^adx[/itex]?

  1. Feb 17, 2014 #1
    Hello I am trying to find a normalizing factor for
    [itex]\int^{3.5}_{0.5}(sin(x*π)+1)^1dx[/itex]

    I want that the integral of [itex]\int^{3.5}_{0.5}(sin(x*π)+1)^a dx *factor = \int^{3.5}_{0.5}(sin(x*π)+1)^1dx[/itex]

    So for this I need to solve the integral
    [itex]\int^{3.5}_{0.5}(sin(x*π)+1)^adx[/itex]

    But I did not mange to solve it.
    I did found that when I do = a = [1,2,3,4,5,6,7]
    I find in wolfram this:
    http://www.wolframalpha.com/input/?i=2+3+5+8.75+15.75+28.875

    in the end of the page we can see a possible sequenct identification:
    [itex]α_{n} = \frac{2^{n}* (\frac{3}{2} )_{n-1} }{(2)_{n-1}}[/itex]

    when σ_n is the Pochhammer symbol.

    Thank You.
     
  2. jcsd
  3. Feb 17, 2014 #2

    DrClaude

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    Staff: Mentor

    Here is what Mathematica gives:
    $$
    \int_{\frac{1}{2}}^{\frac{7}{2}} (\sin (\pi x)+1)^a \, dx =
    \frac{2^a \left(3 \sqrt{\pi } \Gamma \left(a+\frac{1}{2}\right)-2 \Gamma (a+1) B_0\left(a+\frac{1}{2},\frac{1}{2}\right)\right)}{\pi \Gamma (a+1)}
    $$
    provided ##\Re(a) > -1/2##. ##\Gamma## is the gamma function and ##B_0## is the incomplete beta function.
     
  4. Feb 17, 2014 #3

    D H

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    Staff Emeritus
    Science Advisor

    If appears that you are only worried about a being a positive integer. If that's the case, denote ##k_a \equiv \int_{1/2}^{7/2} (1+\sin(\pi x))^a\,dx##. Expand this as ##k_a = \int_{1/2}^{7/2} (1+\sin(\pi x))^{a-1}\,dx + \int_{1/2}^{7/2} (1+\sin(\pi x))^{a-1}\sin(\pi x)\,dx = k_{a-1} + I_a## where ##I_a \equiv \int_{1/2}^{7/2} (1+\sin(\pi x))^{a-1}\sin(\pi x)\,dx##. Integrate this by parts. You should get a linear combination of ##k_{a-1}## and ##k_a##. Solve ##k_a = k_{a-1}+I_a## for ##k_a##. This will give a recursive relation for ##k_a## in terms of ##k_{a-1}##. You know ##k_1##, so this will let you solve for ##k_a## in terms of ##a##.
     
  5. Feb 17, 2014 #4
    Thank You.
    Do you, but this will not help me, Because in your solution, I need to calculate the gamma function, and this is also an integral. I can do it computationally .. But I can just calculate the first integral.


    Thank You D H.. But I also a=0.1 0.5 1.5 ... I am also worried about a being not integral ( but I do not care if it is not positive ).
    Your solution will help me for integer a. :-)
     
  6. Feb 17, 2014 #5
  7. Feb 17, 2014 #6
    Thank You, I write a program that should normalize the integral for every α. and I write it in a wrapper for c.. all I have are those functions:
    http://www.neuron.yale.edu/neuron/static/docs/nmodl/nmodlfunc.html

    I do not have the gamma function and incomplete beta function.
     
  8. Feb 17, 2014 #7

    DrClaude

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    Staff: Mentor

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