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Help with integration

  1. Oct 2, 2009 #1
    1. The problem statement, all variables and given/known data
    Evaluate and give an exact answer.

    [tex]$\int_{0}^{\frac{Pi}{2}\right)}cos^2(2*Pi*n*x)dx[/tex] where n is a positive integer.


    2. Relevant equations



    3. The attempt at a solution
    [tex](change)x=\frac{b-a}{n}\right)=\frac{Pi}{2n}\right)[/tex]
    and...
    [tex]x_i=a+i(change)x[/tex]

    I'm not sure how to write a Riemann Sum in latex code, so I'll be using "S" as the notation for the Riemann Sum (i=1, n): f(x_i)*(b-a)/n

    I used the limit as n -> oo definition of an integral:

    [tex]$\int_{0}^{\frac{Pi}{2}\right)}cos^2(2*Pi*n*x)dx=lim(n->oo)S[f(x_i)(\frac{b-a}{n}\right))][/tex]

    Next I substituted the values of [tex]x_i[/tex]:

    [tex]lim(n->oo)S[f(\frac{ib}{n}\right))(\frac{Pi}{2n}\right))][/tex]

    Then evaluated [tex]f(x_i)[/tex]:

    [tex]lim(n->oo)S[cos^2(2*Pi*n*\frac{Pi}{2n}\right)][/tex]

    Here is where i get a bit confused. In the example with n-subintervals that I worked through in my book, it substituted the value of the Riemann sum of i using the power of sums formula, however, I end up with "i" inside of the cosine function, and am not sure how to finish evaluating the integral.

    Other examples in the book had the value of n defined, thus making the integration process much more simple.

    Am I even doing the right thing? Bare with me, my integration is self taught.
     
  2. jcsd
  3. Oct 2, 2009 #2

    HallsofIvy

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    Staff Emeritus
    Science Advisor

    Why are you trying to change to Riemann sums? That is, at best, a very difficult way of evaluating integrals! (Though excellent for thinking how to set up integrals for applications.)

    Do you know the anti-derivatives of sine and cosine themselves? If so you can use the trig identity [itex]cos^2(\theta)= (1/2)(1+ cos(2\theta)[/itex].
     
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