- #1
Samuelb88
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Homework Statement
Evaluate and give an exact answer.
[tex]$\int_{0}^{\frac{Pi}{2}\right)}cos^2(2*Pi*n*x)dx[/tex] where n is a positive integer.
Homework Equations
The Attempt at a Solution
[tex](change)x=\frac{b-a}{n}\right)=\frac{Pi}{2n}\right)[/tex]
and...
[tex]x_i=a+i(change)x[/tex]
I'm not sure how to write a Riemann Sum in latex code, so I'll be using "S" as the notation for the Riemann Sum (i=1, n): f(x_i)*(b-a)/n
I used the limit as n -> oo definition of an integral:
[tex]$\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.