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To evaluate the integral numerically using Beta and Gamma functions, let's proceed step by step: ### Given Integral: \[ I = \int_{0}^{4} \frac{dx}{\sqrt{4x - x^2}} \] #### Step 1: Convert the Integral into a Standard Beta Function Form We complete the square in the denominator: \[ 4x - x^2 = 4 - (x-2)^2. \] Let: \[ x = 2 + 2\cos\theta, \quad dx = -2\sin\theta \, d\theta. _________ _______ ________ ________ ___ ____ _____ ______ ____.
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