Examine an approximation value of 𝑙𝑜𝑔5 by calculating to 4 
decimal place by Simpsons’s 1
8
rule ∫
𝑑𝑥
4𝑥+5
5
0
dividing the range into 
10 equal parts

To approximate the value of \(\log 5\) using Simpson's \(\frac{1}{8}\) rule for the integral \(\int_0^5 \frac{dx}{4x+5}\) by dividing the range into 10 equal parts, we follow these steps:

### Step 1: Define the Function and Intervals
The integral we need to approximate is:

\[
\int_0^5 \frac{dx}{4x+5}
\]

We divide the interval \([0, 5]\) into 10 equal parts. Therefore, the width of each subinterval \(h\) is:

\[
h = \frac{5 - 0}{10} = 0.5
\]

### Step 2: Calculate the Function Values
We calculate the function values at each point. The function is \(f(x) = \frac{1}{4x+5}\).

Calculate \(f(x_i)\) at \(x_i = 0, 0.5, 1.0, 1.5, 2.0, 2.5, 3.0, 3.5, 4.0, 4.5, 5.0\).

### Step 3: Apply Simpson's \(\frac{1}{8}\) Rule
Simpson's \(\frac{1}{8}\) rule is given by:

\[
\int_{a}^{b} f(x) \, dx \approx \frac{h}{8} [f(x_0) + 3f(x_1) + 3f(x_2) + 2f(x_3) + 3f(x_4) + 3f(x_5) + 2f(x_6) + 3f(x_7) + 3f(x_8) + 2f(x_9) + f(x_{10})]
\]

### Step 4: Calculate Each Term
1. \(f(x_0) = f(0) = \frac{1}{4(0)+5} = \frac{1}{5} = 0.2\)
2. \(f(x_1) = f(0.5) = \frac{1}{4(0.5)+5} = \frac{1}{7} \approx 0.142857\)
3. \(f(x_2) = f(1.0) = \frac{1}{4(1)+5} = \frac{1}{9} \approx 0.111111\)
4. \(f(x_3) = f(1.5) = \frac{1}{4(1.5)+5} = \frac{1}{11} \approx 0.090909\)
5. \(f(x_4) = f(2.0) = \frac{1}{4(2)+5} = \frac{1}{13} \approx 0.076923\)
6. \(f(x_5) = f(2.5) = \frac{1}{4(2.5)+5} = \frac{1}{15} \approx 0.066667\)
7. \(f(x_6) = f(3.0) = \frac{1}{4(3)+5} = \frac{1}{17} \approx 0.058824\)
8. \(f(x_7) = f(3.5) = \frac{1}{4(3.5)+5} = \frac{1}{19} \approx 0.052632\)
9. \(f(x_8) = f(4.0) = \frac{1}{4(4)+5} = \frac{1}{21} \approx 0.047619\)
10. \(f(x_9) = f(4.5) = \frac{1}{4(4.5)+5} = \frac{1}{23} \approx 0.043478\)
11. \(f(x_{10}) = f(5.0) = \frac{1}{4(5)+5} = \frac{1}{25} = 0.04\)

### Step 5: Calculate the Integral Approximation
Combine the function values with the coefficients from Simpson's \(\frac{1}{8}\) rule:

\[
\begin{aligned}
\int_0^5 \frac{dx}{4x+5} &\approx \frac{0.5}{8} \left[0.2 + 3(0.142857) + 3(0.111111) + 2(0.090909) + 3(0.076923) + 3(0.066667) + 2(0.058824) + 3(0.052632) + 3(0.047619) + 2(0.043478) + 0.04\right] \\
&= \frac{0.5}{8} \left[0.2 + 0.428571 + 0.333333 + 0.181818 + 0.230769 + 0.2 + 0.117648 + 0.157896 + 0.142857 + 0.086956 + 0.04\right] \\
&= \frac{0.5}{8} \left[2.120848\right] \\
&= \frac{0.5}{8} \times 2.120848 \\
&= 0.132553
\end{aligned}
\]

So, the approximate value of \(\int_0^5 \frac{dx}{4x+5}\) using Simpson's \(\frac{1}{8}\) rule is \(\boxed{0.1326}\) (to four decimal places).

Would you like more details or have any questions about this process? Here are some related questions you might be interested in:

1. How is Simpson's \(\frac{1}{3}\) rule different from Simpson's \(\frac{1}{8}\) rule?
2. How do we derive Simpson's rule?
3. Can you explain the trapezoidal rule and compare it with Simpson's rule?
4. How do numerical integration methods compare with analytical integration?
5. What are some real-world applications of numerical integration?

**Tip:** When using numerical methods for integration, always consider the number of subintervals and their effect on accuracy. More subintervals generally yield more accurate results.

Learn how to approximate the value of \(\log 5\) using Simpson's \(\frac{1}{8}\) rule for the integral \(\int_0^5 \frac{dx}{4x+5}\), dividing the range into 10 equal parts. Follow detailed steps and calculations to achieve an accurate approximation.

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