determine the following integral s(sin^2 (2X) + cos^2 (2x)) (x^2 - 2cos pi + 2/x^2 )dx

Note it is often necessary to simplify the integrand first
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The integrand simplifies; cos((pi))=-1 and the sum of the squares of the sine and cosine of any angle=1, so the integrand becomes x^2+2+2x^-2. The integral is x^3/3+2x-2/x+c where c is constant of integration.

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