Word Count : 361

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First, we evaluate the integral [ \int_0^1 \frac{1}{1+x^2} , dx ] using the trapezoidal rule. Step 1: Trapezoidal Rule with (h = 0.5) The formula: [ T = \frac{h}{2} \big[f(x_0) + 2f(x_1) + f(x_2)\big] ] Divide [0,1] with (h=0.5): (x_0=0, x_1=0.5, x_2=1) [ f(0) = \frac{1}{1+0^2} = 1 ] [ f(0.5) = \frac{1}{1+0.5^2} = \frac{1}{1.25} = 0.8 ] [ f(1) = \frac{1}{1+1^2} = \frac{1}{2} = 0.5 ] [ T_{0.5} = \frac{0.5}{2} \big[1 + 2(0.8) + 0.5\big] = 0.25 \cdot (1 + 1.6 + 0.5) = 0.25 \cdot 3.1 = 0.775 ___ _________ _________ ________ _________ ________ _________ ______ _______ ____ ____ _______.
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