Word Count : 482

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We are asked to approximate the integral $$ \int_0^1 \frac{dx}{1+x^2} $$ using composite Simpson's rule with $h = 0.25$ and $h = 0.125$, and then improve the result using Romberg Integration. --- ### Step 1: Define the function Let $$ f(x) = \frac{1}{1+x^2} $$ --- ## Using Composite Simpson’s Rule Simpson’s Rule (composite form): For $n$ even, step size $h = \frac{b-a}{n}$, the rule is: $$ \int_a^b f(x) \, dx \approx \frac{h}{3} \left[ f(x_0) + 4 \sum_{\text{odd } i} f(x_i) + 2 \sum_{\text{even } i \ne 0,n} f(x_i) + f(x_n) \right] $$ --- ### Case 1: __________ _____ ________ ______ ____ _____ ____ ____ ______ _______.
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