Word Count : 271

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We will solve the given integral manually, step by step: \[ I = \int_{0}^{2\pi} \frac{d\theta}{2 + \cos\theta} \] ### Step 1: Use the Weierstrass Substitution We use the substitution: \[ \tan\frac{\theta}{2} = t \] This substitution transforms trigonometric functions as: \[ d\theta = \frac{2\, dt}{1 + t^2} \] \[ \cos\theta = \frac{1 - t^2}{1 + t^2} \] For the limits: - When \( \theta = 0 \Rightarrow ______ _____ _____ _________ ______ _______ ______.
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