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Need help in deriving this reduction formula

  1. Jan 8, 2012 #1
    It might be difficult for you to read this integral in non latex form, but i'll try my best.
    As i dont know how to write this in latex form, assume "for this problem" that I(n) is pronounced as "I subscript n" or nth term of I.


    1. The problem statement, all variables and given/known data

    I(n)=∫ (sinx)^n dx [with limits of the integral as : from zero (0) to ∏/2 (pi/2)

    Using the above equation, it is required to prove that :

    I(n+2)= I(n) * (n+1)/(n+2) [again , I(n) means I subscript n ie nth term of a sequence]
    2. Relevant equations
    the formula for integration by parts


    3. The attempt at a solution

    I have tried to integrate it by parts using various ways but all of them failed to prove the required result.
    One of them was :
    ∫ [sin^-2(x)*(sin(x))^(n+2)] dx [with the same limits ofcourse]

    even after subsituting 1-cos^2(x) for sin^2(x) the problem could not be solved,
    please help me solve it.
     
  2. jcsd
  3. Jan 8, 2012 #2
  4. Jan 8, 2012 #3

    SammyS

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    Start with In+2.
    Even without LaTeX, you can do SUBSCRIPTS and SUPERSCRIPTS by using the X2 and X2 buttons in the 'Go Advanced' message window.​
    [itex]\displaystyle I_{n+2}=\int\sin^{n+2}(x)\,dx=\int(1-\cos^2(x))\sin^{n}(x)\,dx=I_n-\int\cos^2(x)\sin^{n}(x)\,dx[/itex]

    Evaluate that last integral using integration by parts.
    u=cos(x), dv=cos(x)sinn(x) dx ​
     
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