Answer
$$\eqalign{
& {\text{Trapezoidal Rule}} \approx 0.28483 \cr
& {\text{Simpson's Rule}} \approx 0.28380 \cr
& {\text{Graphing utility}} \approx 0.28379 \cr} $$
Work Step by Step
$$\eqalign{
& \int_2^3 {\frac{2}{{1 + {x^2}}}} dx,{\text{ then }}\underbrace {\left[ {2,3} \right]}_{\left[ {a,b} \right]} \cr
& {\text{For }}n = 4,{\text{ }}\Delta x = \frac{{b - a}}{n} = \frac{{3 - 2}}{4} = \frac{1}{4},{\text{ then,}} \cr
& \cr
& {\text{*Using the trapezoidal Rule }}\left( {{\text{THEOREM 4}}{\text{.17}}} \right) \cr
& \int_a^b {f\left( x \right)} dx \approx \frac{{b - a}}{{2n}}\left[ {f\left( {{x_0}} \right) + 2f\left( {{x_1}} \right) + \cdots 2f\left( {{x_{n - 1}}} \right) + f\left( {{x_n}} \right)} \right] \cr
& {x_0} = 2,{\text{ }}{x_1} = \frac{9}{4},{\text{ }}{x_2}{\text{ = }}\frac{5}{2}{\text{, }}{x_3} = \frac{{11}}{4},{\text{ }}{x_4} = 3 \cr
& \cr
& \int_2^3 {\frac{2}{{1 + {x^2}}}} dx \approx \frac{1}{{2\left( 4 \right)}}\left[ {f\left( 2 \right) + 2f\left( {\frac{9}{4}} \right) + 2f\left( {\frac{5}{2}} \right)} \right] \cr
& + \frac{1}{{2\left( 4 \right)}}\left[ {2f\left( {\frac{{11}}{4}} \right) + f\left( 3 \right)} \right] \cr
& \int_2^3 {\frac{2}{{1 + {x^2}}}} dx \approx \frac{1}{8}\left[ {\frac{2}{{1 + {{\left( 2 \right)}^2}}} + \frac{4}{{1 + {{\left( {9/4} \right)}^2}}} + \frac{4}{{1 + {{\left( {5/2} \right)}^2}}}} \right] \cr
& {\text{ }} + \frac{1}{8}\left[ {\frac{4}{{1 + {{\left( {11/4} \right)}^2}}} + \frac{2}{{1 + {{\left( 3 \right)}^2}}}} \right] \cr
& {\text{Simplifying}} \cr
& \int_2^3 {\frac{2}{{1 + {x^2}}}} dx \approx 0.28483 \cr
& \cr
& {\text{*Using the Simpson's Rule }}\left( {{\text{THEOREM 4}}{\text{.19}}} \right) \cr
& \int_a^b {f\left( x \right)} dx \approx \frac{{b - a}}{{3n}}\left[ {f\left( {{x_0}} \right) + 4f\left( {{x_1}} \right) + 2f\left( {{x_2}} \right) + 4f\left( {{x_3}} \right) + \cdots } \right. \cr
& \left. { + 4f\left( {{x_{n - 1}}} \right) + f\left( {{x_n}} \right)} \right] \cr
& \int_2^3 {\frac{2}{{1 + {x^2}}}} dx \approx \frac{1}{{3\left( 4 \right)}}\left[ {f\left( 2 \right) + 4f\left( {\frac{9}{4}} \right) + 2f\left( {\frac{5}{2}} \right)} \right] \cr
& + \frac{1}{{3\left( 4 \right)}}\left[ {4f\left( {\frac{{11}}{4}} \right) + f\left( 3 \right)} \right] \cr
& \int_2^3 {\frac{2}{{1 + {x^2}}}} dx \approx \frac{1}{{12}}\left[ {\frac{2}{{1 + {{\left( 2 \right)}^2}}} + \frac{8}{{1 + {{\left( {9/4} \right)}^2}}} + \frac{4}{{1 + {{\left( {5/2} \right)}^2}}}} \right] \cr
& {\text{ }} + \frac{1}{{12}}\left[ {\frac{8}{{1 + {{\left( {11/4} \right)}^2}}} + \frac{2}{{1 + {{\left( 3 \right)}^2}}}} \right] \cr
& {\text{Simplifying}} \cr
& \int_2^3 {\frac{2}{{1 + {x^2}}}} dx \approx 0.28380 \cr
& \cr
& {\text{Using a graphing utility we obtain}} \cr
& \int_2^3 {\frac{2}{{1 + {x^2}}}} dx \approx 0.28379 \cr} $$
