This video demonstrates how to complete an Integral that looks very difficult to Integrate in real analysis, but can be simplified somewhat by substituting into a Complex Integral and applying Cauchy's Residue Theorem .
So doing a substitution cos x to 1/2(z+1/z) and dx to 1/iz dz.
Then substituting the terms and manipulate the Algebra to get a function with Polynomials.
Then find the residues of the function .
Then the parametric transformation of the parameters of integration are transformed to a Closed Contour we call Gamma.
The residues that are within the Gamma are then summed and multiplied by 2 pi i .
The answer should always be a real value as we started with a Real Value Integral.
https://youtu.be/mNxMubJop80
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