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To evaluate the integral using Cauchy's residue theorem, we first express the integral in a complex form. The given integral is: \[ I = \int_{0}^{2\pi} \frac{d\theta}{2 + \cos\theta} \] ### Step 1: Convert to a complex integral We use the substitution \( z = e^{i\theta} \), so that \( dz = i e^{i\theta} d\theta = i z d\theta \). Also, note that \( \cos\theta = \frac{z + z^{-1}}{2} \). Thus, the integral becomes: \[ I = \int_{|z|=1} \frac{dz}{i z \left( 2 ________ ______ __________ ____ ____ ____ _________ __________ ____ _____.
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