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Sagot :
To evaluate the integral [tex]\(\int_0^5 \frac{dx}{4x+5}\)[/tex] using Simpson's [tex]\(\left(\frac{1}{3}\right)^{rd}\)[/tex] rule with 11 ordinates, follow these steps:
1. Given Parameters:
- Function: [tex]\(f(x) = \frac{1}{4x + 5}\)[/tex]
- Lower limit of integration [tex]\(a = 0\)[/tex]
- Upper limit of integration [tex]\(b = 5\)[/tex]
- Number of ordinates = 11, meaning there are 10 intervals ([tex]\(n = 10\)[/tex])
2. Calculate the width of each interval ([tex]\(h\)[/tex]):
[tex]\[ h = \frac{b - a}{n} = \frac{5 - 0}{10} = 0.5 \][/tex]
3. Apply Simpson's [tex]\(\frac{1}{3}\)[/tex] rule:
The integral can be approximated using Simpson's rule as follows:
[tex]\[ \int_{a}^{b} f(x) \, dx \approx \frac{h}{3} \left[ f(a) + f(b) + 4 \sum_{\text{odd } i} f(x_i) + 2 \sum_{\text{even } i} f(x_i) \right] \][/tex]
4. Calculate the values of [tex]\(f(x)\)[/tex] at the required points:
- [tex]\(f(x_0) = f(0) = \frac{1}{4(0) + 5} = \frac{1}{5}\)[/tex]
- [tex]\(f(x_{10}) = f(5) = \frac{1}{4(5) + 5} = \frac{1}{25}\)[/tex]
5. Compute the terms for odd and even indices:
- For [tex]\(x_1, x_3, x_5, x_7, x_9\)[/tex] (odd indices):
[tex]\[ x_i = a + i \cdot h \][/tex]
[tex]\[ f(x_1) = f(0.5) = \frac{1}{4(0.5) + 5} = \frac{1}{7}, \quad f(x_3) = f(1.5) = \frac{1}{11}, \quad f(x_5) = f(2.5) = \frac{1}{15}, \quad f(x_7) = f(3.5) = \frac{1}{19}, \quad f(x_9) = f(4.5) = \frac{1}{23} \][/tex]
The sum for odd indices:
[tex]\[ 4 \left( f(x_1) + f(x_3) + f(x_5) + f(x_7) + f(x_9) \right) \][/tex]
- For [tex]\(x_2, x_4, x_6, x_8\)[/tex] (even indices):
[tex]\[ x_i = a + i \cdot h \][/tex]
[tex]\[ f(x_2) = f(1) = \frac{1}{9}, \quad f(x_4) = f(2) = \frac{1}{13}, \quad f(x_6) = f(3) = \frac{1}{17}, \quad f(x_8) = f(4) = \frac{1}{21} \][/tex]
The sum for even indices:
[tex]\[ 2 \left( f(x_2) + f(x_4) + f(x_6) + f(x_8) \right) \][/tex]
6. Combine all terms to find the integral:
[tex]\[ \int_0^5 \frac{dx}{4x+5} \approx \frac{h}{3} \left[ f(a) + f(b) + 4 \left( f(x_1) + f(x_3) + f(x_5) + f(x_7) + f(x_9) \right) + 2 \left( f(x_2) + f(x_4) + f(x_6) + f(x_8) \right) \right] \][/tex]
7. Evaluate the above expression:
[tex]\[ \frac{0.5}{3} \left[ \frac{1}{5} + \frac{1}{25} + 4 \left( \frac{1}{7} + \frac{1}{11} + \frac{1}{15} + \frac{1}{19} + \frac{1}{23} \right) + 2 \left( \frac{1}{9} + \frac{1}{13} + \frac{1}{17} + \frac{1}{21} \right) \right] \][/tex]
8. Final Result:
[tex]\[ \approx 0.40252074852155617 \][/tex]
Thus, the approximation of the integral [tex]\(\int_0^5 \frac{dx}{4x+5}\)[/tex] using Simpson's [tex]\(\left(\frac{1}{3}\right)^{rd}\)[/tex] rule with 11 ordinates is approximately [tex]\(0.40252074852155617\)[/tex].
1. Given Parameters:
- Function: [tex]\(f(x) = \frac{1}{4x + 5}\)[/tex]
- Lower limit of integration [tex]\(a = 0\)[/tex]
- Upper limit of integration [tex]\(b = 5\)[/tex]
- Number of ordinates = 11, meaning there are 10 intervals ([tex]\(n = 10\)[/tex])
2. Calculate the width of each interval ([tex]\(h\)[/tex]):
[tex]\[ h = \frac{b - a}{n} = \frac{5 - 0}{10} = 0.5 \][/tex]
3. Apply Simpson's [tex]\(\frac{1}{3}\)[/tex] rule:
The integral can be approximated using Simpson's rule as follows:
[tex]\[ \int_{a}^{b} f(x) \, dx \approx \frac{h}{3} \left[ f(a) + f(b) + 4 \sum_{\text{odd } i} f(x_i) + 2 \sum_{\text{even } i} f(x_i) \right] \][/tex]
4. Calculate the values of [tex]\(f(x)\)[/tex] at the required points:
- [tex]\(f(x_0) = f(0) = \frac{1}{4(0) + 5} = \frac{1}{5}\)[/tex]
- [tex]\(f(x_{10}) = f(5) = \frac{1}{4(5) + 5} = \frac{1}{25}\)[/tex]
5. Compute the terms for odd and even indices:
- For [tex]\(x_1, x_3, x_5, x_7, x_9\)[/tex] (odd indices):
[tex]\[ x_i = a + i \cdot h \][/tex]
[tex]\[ f(x_1) = f(0.5) = \frac{1}{4(0.5) + 5} = \frac{1}{7}, \quad f(x_3) = f(1.5) = \frac{1}{11}, \quad f(x_5) = f(2.5) = \frac{1}{15}, \quad f(x_7) = f(3.5) = \frac{1}{19}, \quad f(x_9) = f(4.5) = \frac{1}{23} \][/tex]
The sum for odd indices:
[tex]\[ 4 \left( f(x_1) + f(x_3) + f(x_5) + f(x_7) + f(x_9) \right) \][/tex]
- For [tex]\(x_2, x_4, x_6, x_8\)[/tex] (even indices):
[tex]\[ x_i = a + i \cdot h \][/tex]
[tex]\[ f(x_2) = f(1) = \frac{1}{9}, \quad f(x_4) = f(2) = \frac{1}{13}, \quad f(x_6) = f(3) = \frac{1}{17}, \quad f(x_8) = f(4) = \frac{1}{21} \][/tex]
The sum for even indices:
[tex]\[ 2 \left( f(x_2) + f(x_4) + f(x_6) + f(x_8) \right) \][/tex]
6. Combine all terms to find the integral:
[tex]\[ \int_0^5 \frac{dx}{4x+5} \approx \frac{h}{3} \left[ f(a) + f(b) + 4 \left( f(x_1) + f(x_3) + f(x_5) + f(x_7) + f(x_9) \right) + 2 \left( f(x_2) + f(x_4) + f(x_6) + f(x_8) \right) \right] \][/tex]
7. Evaluate the above expression:
[tex]\[ \frac{0.5}{3} \left[ \frac{1}{5} + \frac{1}{25} + 4 \left( \frac{1}{7} + \frac{1}{11} + \frac{1}{15} + \frac{1}{19} + \frac{1}{23} \right) + 2 \left( \frac{1}{9} + \frac{1}{13} + \frac{1}{17} + \frac{1}{21} \right) \right] \][/tex]
8. Final Result:
[tex]\[ \approx 0.40252074852155617 \][/tex]
Thus, the approximation of the integral [tex]\(\int_0^5 \frac{dx}{4x+5}\)[/tex] using Simpson's [tex]\(\left(\frac{1}{3}\right)^{rd}\)[/tex] rule with 11 ordinates is approximately [tex]\(0.40252074852155617\)[/tex].
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