Get detailed and reliable answers to your questions on IDNLearn.com. Our platform is designed to provide accurate and comprehensive answers to any questions you may have.

Apply the Trapezoidal Rule, let [tex]n=4[/tex].

When [tex]n=4[/tex], [tex]\Delta x=\square[/tex], therefore the corresponding [tex]x[/tex] values are the following:
[tex]\[
\begin{array}{l}
x_0=\square \\
x_1=\square \\
x_2=\square \\
x_3=\square \\
x_4=1
\end{array}
\][/tex]

Substitute these values in the Trapezoidal Rule:
[tex]\[
\int_0^1 \frac{2}{(x+2)^2} \, dx \approx \frac{1}{8}\left[\frac{2}{(\square+2)^2} + 2\left(\frac{2}{(\square+2)^2}\right) + 2\left(\frac{2}{(\square+2)^2}\right) + 2\left(\frac{2}{(\square+2)^2}\right) + \left(\frac{2}{(\square+2)^2}\right)\right]
\][/tex]

[tex]\[
\approx \frac{1}{8} \square + 2\left(\frac{\square}{81}\right) + 2\left(\frac{\square}{25}\right)
\][/tex]


Sagot :

Let's apply the Trapezoidal Rule with the given values step by step.

#### Step 1: Determine [tex]\(\Delta x\)[/tex]
For [tex]\(n=4\)[/tex], the interval [tex]\([0, 1]\)[/tex] is divided into 4 subintervals.
[tex]\[ \Delta x = \frac{1-0}{4} = 0.25 \][/tex]

#### Step 2: Calculate [tex]\(x\)[/tex] values
The [tex]\(x\)[/tex] values are calculated as follows:
[tex]\[ \begin{array}{l} x_0 = 0 \cdot 0.25 = 0 \\ x_1 = 1 \cdot 0.25 = 0.25 \\ x_2 = 2 \cdot 0.25 = 0.5 \\ x_3 = 3 \cdot 0.25 = 0.75 \\ x_4 = 4 \cdot 0.25 = 1 \\ \end{array} \][/tex]

#### Step 3: Formulate the Trapezoidal Rule
Given the function [tex]\( f(x) = \frac{2}{(x+2)^2} \)[/tex], we can apply the Trapezoidal Rule:
[tex]\[ \int_0^1 \frac{2}{(x+2)^2} \, dx \approx \frac{\Delta x}{2} \left[ f(x_0) + 2f(x_1) + 2f(x_2) + 2f(x_3) + f(x_4) \right] \][/tex]
Substituting the [tex]\(x\)[/tex] values:
[tex]\[ \begin{array}{l} \int_0^1 \frac{2}{(x+2)^2} \, dx \approx \frac{0.25}{2} \left[ \frac{2}{(0+2)^2} \right. \\ + 2 \left( \frac{2}{(0.25+2)^2} \right) \\ + 2 \left( \frac{2}{(0.5+2)^2} \right) \\ + 2 \left( \frac{2}{(0.75+2)^2} \right) \\ \left. + \frac{2}{(1+2)^2} \right] \end{array} \][/tex]

#### Step 4: Evaluate the function at the [tex]\(x\)[/tex] values
[tex]\[ \begin{array}{l} f(x_0) = \frac{2}{(0+2)^2} = \frac{2}{4} = 0.5 \\ f(x_1) = \frac{2}{(0.25+2)^2} = \frac{2}{5.0625} \approx 0.394 \\ f(x_2) = \frac{2}{(0.5+2)^2} = \frac{2}{6.25} = 0.32 \\ f(x_3) = \frac{2}{(0.75+2)^2} = \frac{2}{7.5625} \approx 0.264 \\ f(x_4) = \frac{2}{(1+2)^2} = \frac{2}{9} \approx 0.222 \\ \end{array} \][/tex]

#### Step 5: Substitute and compute
[tex]\[ \begin{array}{l} \int_0^1 \frac{2}{(x+2)^2} \, dx \approx \frac{0.25}{2} \left[0.5 \right.\\ + 2 \left(0.394\right) + 2 \left(0.32\right) + 2 \left(0.264\right) + 0.222 \\ \left.\right] \\ \approx \frac{0.25}{2} \left[0.5 + 0.788 + 0.64 + 0.528 + 0.222\right] \\ = \frac{0.25}{2} \left[2.678\right] \\ = 0.125 \cdot 2.678 \\ \approx 0.335 \end{array} \][/tex]

Hence, we successfully calculated the integral [tex]\(\int_0^1 \frac{2}{(x+2)^2} \, dx \)[/tex] using the Trapezoidal Rule with an approximate value of 0.335.
We appreciate your participation in this forum. Keep exploring, asking questions, and sharing your insights with the community. Together, we can find the best solutions. For dependable and accurate answers, visit IDNLearn.com. Thanks for visiting, and see you next time for more helpful information.