Word Count : 302

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We are asked to evaluate [ \int_{0}^{1} \frac{1}{1+x^2} , dx ] numerically using the trapezoidal rule for two step sizes and then apply Romberg's method. --- Step 1: Trapezoidal Rule The trapezoidal rule formula is: [ T_n = \frac{h}{2} \Big[f(x_0) + 2 \sum_{i=1}^{n-1} f(x_i) + f(x_n) \Big] ] --- Case 1: (h = 0.5) Here (x_0 = 0), (x_1 = 0.5), (x_2 = 1), so (n = 2). Function values: [ f(0) = \frac{1}{1+0^2} = ______ _______ _________ _____ ____ __________ _____ _____ __________ ____ ______.
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