Chapter 4 - Integration - 4.6 Exercises - Page 310: 19

$$\eqalign{ & {\text{Trapezoidal Rule}} \approx 0.1940 \cr & {\text{Simpson's Rule}} \approx 0.18596 \cr & {\text{Graphing utility}} \approx 0.18578 \cr} $$

$$\eqalign{ & \int_0^{\pi /4} {x\tan x} dx \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 & {\text{For }}n = 4,{\text{ }}\Delta x = \frac{{b - a}}{n} = \frac{{\pi /4 - 0}}{4} = \frac{\pi }{{16}},{\text{ then,}} \cr & {x_0} = 0,{\text{ }}{x_1} = \frac{\pi }{{16}},{\text{ }}{x_2} = \frac{\pi }{8}{\text{, }}{x_3} = \frac{{3\pi }}{{16}},{\text{ }}{x_4} = \frac{\pi }{4} \cr & f\left( {{x_0}} \right) = f\left( 0 \right) = 0\tan 0 = 0 \cr & f\left( {{x_1}} \right) = f\left( {\frac{\pi }{{16}}} \right) = \frac{\pi }{{16}}\tan \left( {\frac{\pi }{{16}}} \right) \cr & f\left( {{x_2}} \right) = f\left( {\frac{\pi }{8}} \right) = \frac{\pi }{8}\tan \left( {\frac{\pi }{8}} \right) \cr & f\left( {{x_3}} \right) = f\left( {\frac{{3\pi }}{{16}}} \right) = \frac{{3\pi }}{{16}}\tan \left( {\frac{{3\pi }}{{16}}} \right) \cr & f\left( {{x_4}} \right) = f\left( {\frac{\pi }{4}} \right) = \frac{\pi }{4}\tan \left( {\frac{\pi }{4}} \right) = \frac{\pi }{4} \cr & {\text{Therefore,}} \cr & \int_0^{\pi /4} {x\tan x} dx \approx \frac{\pi }{{2\left( {16} \right)}}\left[ {0 + \frac{{2\pi }}{{16}}\tan \left( {\frac{\pi }{{16}}} \right) + \frac{{2\pi }}{8}\tan \left( {\frac{\pi }{8}} \right)} \right] \cr & + \frac{\pi }{{2\left( {16} \right)}}\left[ {2\left( {\frac{{3\pi }}{{16}}} \right)\tan \left( {\frac{{3\pi }}{{16}}} \right) + \frac{\pi }{4}} \right] \cr & {\text{Simplifying by using a calculator}} \cr & \int_0^{\pi /4} {x\tan x} dx \approx 0.1940 \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_0^{\pi /4} {x\tan x} dx \approx \frac{\pi }{{3\left( {16} \right)}}\left[ {0 + \frac{{4\pi }}{{16}}\tan \left( {\frac{\pi }{{16}}} \right) + \frac{{2\pi }}{8}\tan \left( {\frac{\pi }{8}} \right)} \right] \cr & + \frac{\pi }{{3\left( {16} \right)}}\left[ {4\left( {\frac{{3\pi }}{{16}}} \right)\tan \left( {\frac{{3\pi }}{{16}}} \right) + \frac{\pi }{4}} \right] \cr & {\text{Simplifying by using a calculator}} \cr & \int_0^{\pi /4} {x\tan x} dx \approx 0.18596 \cr & \cr & {\text{Using a graphing utility we obtain}} \cr & \int_0^{\pi /4} {x\tan x} dx \approx 0.18578 \cr} $$