Answer
$$\eqalign{
& {\text{Trapezoidal Rule}} \approx 0.1719 \cr
& {\text{Simpson's Rule}} \approx 0.1661 \cr
& {\text{Graphing utility}} \approx 0.1657 \cr} $$
Work Step by Step
$$\eqalign{
& \int_0^1 {\frac{{{x^{3/2}}}}{{3 - {x^2}}}} dx \cr
& {\text{For }}n = 4,{\text{ }}\Delta x = \frac{{b - a}}{n} = \frac{{1 - 0}}{4} = \frac{1}{4},{\text{ then,}} \cr
& {x_0} = 0,{\text{ }}{x_1} = \frac{1}{4},{\text{ }}{x_2} = \frac{1}{2},{\text{ }}{x_3} = \frac{3}{4},{\text{ }}{x_4} = 1 \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
& \int_0^1 {\frac{{{x^{3/2}}}}{{3 - {x^2}}}} dx \approx \frac{1}{{2\left( 4 \right)}}\left[ {f\left( 0 \right) + 2f\left( {\frac{1}{4}} \right) + 2f\left( {\frac{1}{2}} \right)} \right] \cr
& + \frac{1}{{2\left( 4 \right)}}\left[ {2f\left( {\frac{3}{4}} \right) + f\left( 1 \right)} \right] \cr
& \int_0^1 {\frac{{{x^{3/2}}}}{{3 - {x^2}}}} dx \approx \frac{1}{8}\left[ {0 + \frac{{2{{\left( {1/4} \right)}^{3/2}}}}{{3 - {{\left( {1/4} \right)}^2}}} + \frac{{2{{\left( {1/2} \right)}^{3/2}}}}{{3 - {{\left( {1/2} \right)}^2}}}} \right] \cr
& {\text{ }} + \frac{1}{8}\left[ {\frac{{2{{\left( {3/4} \right)}^{3/2}}}}{{3 - {{\left( {3/4} \right)}^2}}} + \frac{{{{\left( 1 \right)}^{3/2}}}}{{3 - {{\left( 1 \right)}^2}}}} \right] \cr
& {\text{Simplifying}} \cr
& \int_0^1 {\frac{{{x^{3/2}}}}{{3 - {x^2}}}} dx \approx 0.1719 \cr
& \cr
& {\text{*Using 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_0^1 {\frac{{{x^{3/2}}}}{{3 - {x^2}}}} dx \approx \frac{1}{{3\left( 4 \right)}}\left[ {f\left( 0 \right) + 4f\left( {\frac{1}{4}} \right) + 2f\left( {\frac{1}{2}} \right)} \right] \cr
& + \frac{1}{{3\left( 4 \right)}}\left[ {4f\left( {\frac{3}{4}} \right) + f\left( 1 \right)} \right] \cr
& \int_0^1 {\frac{{{x^{3/2}}}}{{3 - {x^2}}}} dx \approx \frac{1}{{12}}\left[ {0 + \frac{{4{{\left( {1/4} \right)}^{3/2}}}}{{3 - {{\left( {1/4} \right)}^2}}} + \frac{{2{{\left( {1/2} \right)}^{3/2}}}}{{3 - {{\left( {1/2} \right)}^2}}}} \right] \cr
& {\text{ }} + \frac{1}{{12}}\left[ {\frac{{4{{\left( {3/4} \right)}^{3/2}}}}{{3 - {{\left( {3/4} \right)}^2}}} + \frac{{{{\left( 1 \right)}^{3/2}}}}{{3 - {{\left( 1 \right)}^2}}}} \right] \cr
& {\text{Simplifying}} \cr
& \int_2^3 {\frac{2}{{1 + {x^2}}}} dx \approx 0.1661 \cr
& \cr
& {\text{Using a graphing utility we obtain}} \cr
& \int_0^1 {\frac{{{x^{3/2}}}}{{3 - {x^2}}}} dx \approx 0.1657 \cr} $$
