How can I calculate the following integral??

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reza1001gh

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How can I calculate the following integral??



x0 , T are constants

Please help, thank you for your help in advance
 
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reza1001gh, I can't see any text associated to this thread. It seems empty
 

Parameters T and n are constant numbers, so that you should integrate something in the form of \[{ sin(x)}^{4n}\].
Check this pattern here:

 

We know that

sin(a)*sin(b) = cos(a-b)/2 - cos(a+b)/2

if a=(x+xo)*T and b=(x-xo)*T then:

sin[(x+xo)*T]*sin[(x-xo)*T] = cos(2*xo*T)/2-cos(2*x*T)/2

the first term doesn't depends from "x" so it is costant. calling it "k" we will have:

sin[(x+xo)*T]*sin[(x-xo)*T] = [k-cos(2*x*T)]/2

so we will have to integrate:

{[k-cos(2*x*T)]/2}^(2*n)

if "n" is an integer number, the integral can be calculated using the reduction formula (https://en.wikipedia.org/wiki/Integration_by_reduction_formulae). It is quite tedious.
 

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