Answer :
To approximate the given integral
[tex]\[ \int_1^5 \frac{6 \cos (4 x)}{x} \, dx \][/tex]
using the specified methods (a) the Trapezoidal Rule, (b) the Midpoint Rule, and (c) Simpson's Rule with [tex]\( n = 4 \)[/tex], we can follow these steps.
### (a) The Trapezoidal Rule
1. Divide the interval [tex]\([1, 5]\)[/tex] into [tex]\( n = 4 \)[/tex] subintervals. So, the width of each subinterval [tex]\( h \)[/tex] is given by:
[tex]\[ h = \frac{b - a}{n} = \frac{5 - 1}{4} = 1 \][/tex]
2. Determine the partition points [tex]\( x_i \)[/tex] for [tex]\( i = 0 \)[/tex] to [tex]\( n \)[/tex]:
[tex]\[ x_0 = 1, \quad x_1 = 2, \quad x_2 = 3, \quad x_3 = 4, \quad x_4 = 5 \][/tex]
3. Compute the value of the function at each partition point:
[tex]\[ f(x) = \frac{6 \cos (4 x)}{x} \][/tex]
4. Apply the Trapezoidal Rule formula:
[tex]\[ \int_1^5 \frac{6 \cos (4 x)}{x} \, dx \approx \frac{h}{2} \left[ f(x_0) + 2 \sum_{i=1}^{n-1} f(x_i) + f(x_n) \right] \][/tex]
5. The approximate value using the Trapezoidal Rule to six decimal places is:
[tex]\[ -1.901363 \][/tex]
### (b) The Midpoint Rule
1. Divide the interval [tex]\([1, 5]\)[/tex] into [tex]\( n = 4 \)[/tex] subintervals, and the width of each subinterval [tex]\( h \)[/tex] remains:
[tex]\[ h = 1 \][/tex]
2. Determine the midpoint of each subinterval:
[tex]\[ x_{0.5} = 1.5, \quad x_{1.5} = 2.5, \quad x_{2.5} = 3.5, \quad x_{3.5} = 4.5 \][/tex]
3. Compute the value of the function at each midpoint:
[tex]\[ f(x) = \frac{6 \cos (4 x)}{x} \][/tex]
4. Apply the Midpoint Rule formula:
[tex]\[ \int_1^5 \frac{6 \cos (4 x)}{x} \, dx \approx h \sum_{i=1}^{n} f\left( x_{i - 0.5} \right) \][/tex]
5. The approximate value using the Midpoint Rule to six decimal places is:
[tex]\[ 2.941738 \][/tex]
### (c) Simpson's Rule
1. Divide the interval [tex]\([1, 5]\)[/tex] into [tex]\( n = 4 \)[/tex] subintervals of equal width [tex]\( h \)[/tex]:
[tex]\[ h = 1 \][/tex]
2. Compute the value of the function at the partition points and midpoints for alternating coefficients:
[tex]\[ f(x) = \frac{6 \cos (4 x)}{x} \][/tex]
3. Apply Simpson's Rule formula:
[tex]\[ \int_1^5 \frac{6 \cos (4 x)}{x} \, dx \approx \frac{h}{3} \left[ f(x_0) + 4 \sum_{i=1, 3, 5, \cdots} f(x_i) + 2 \sum_{i=2, 4, 6, \cdots (n-2)} f(x_i) + f(x_n) \right] \][/tex]
4. The approximate value using Simpson's Rule to six decimal places is:
[tex]\[ 1.327371 \][/tex]
### Summary of results:
- Trapezoidal Rule: [tex]\(-1.901363\)[/tex]
- Midpoint Rule: [tex]\(2.941738\)[/tex]
- Simpson's Rule: [tex]\(1.327371\)[/tex]
[tex]\[ \int_1^5 \frac{6 \cos (4 x)}{x} \, dx \][/tex]
using the specified methods (a) the Trapezoidal Rule, (b) the Midpoint Rule, and (c) Simpson's Rule with [tex]\( n = 4 \)[/tex], we can follow these steps.
### (a) The Trapezoidal Rule
1. Divide the interval [tex]\([1, 5]\)[/tex] into [tex]\( n = 4 \)[/tex] subintervals. So, the width of each subinterval [tex]\( h \)[/tex] is given by:
[tex]\[ h = \frac{b - a}{n} = \frac{5 - 1}{4} = 1 \][/tex]
2. Determine the partition points [tex]\( x_i \)[/tex] for [tex]\( i = 0 \)[/tex] to [tex]\( n \)[/tex]:
[tex]\[ x_0 = 1, \quad x_1 = 2, \quad x_2 = 3, \quad x_3 = 4, \quad x_4 = 5 \][/tex]
3. Compute the value of the function at each partition point:
[tex]\[ f(x) = \frac{6 \cos (4 x)}{x} \][/tex]
4. Apply the Trapezoidal Rule formula:
[tex]\[ \int_1^5 \frac{6 \cos (4 x)}{x} \, dx \approx \frac{h}{2} \left[ f(x_0) + 2 \sum_{i=1}^{n-1} f(x_i) + f(x_n) \right] \][/tex]
5. The approximate value using the Trapezoidal Rule to six decimal places is:
[tex]\[ -1.901363 \][/tex]
### (b) The Midpoint Rule
1. Divide the interval [tex]\([1, 5]\)[/tex] into [tex]\( n = 4 \)[/tex] subintervals, and the width of each subinterval [tex]\( h \)[/tex] remains:
[tex]\[ h = 1 \][/tex]
2. Determine the midpoint of each subinterval:
[tex]\[ x_{0.5} = 1.5, \quad x_{1.5} = 2.5, \quad x_{2.5} = 3.5, \quad x_{3.5} = 4.5 \][/tex]
3. Compute the value of the function at each midpoint:
[tex]\[ f(x) = \frac{6 \cos (4 x)}{x} \][/tex]
4. Apply the Midpoint Rule formula:
[tex]\[ \int_1^5 \frac{6 \cos (4 x)}{x} \, dx \approx h \sum_{i=1}^{n} f\left( x_{i - 0.5} \right) \][/tex]
5. The approximate value using the Midpoint Rule to six decimal places is:
[tex]\[ 2.941738 \][/tex]
### (c) Simpson's Rule
1. Divide the interval [tex]\([1, 5]\)[/tex] into [tex]\( n = 4 \)[/tex] subintervals of equal width [tex]\( h \)[/tex]:
[tex]\[ h = 1 \][/tex]
2. Compute the value of the function at the partition points and midpoints for alternating coefficients:
[tex]\[ f(x) = \frac{6 \cos (4 x)}{x} \][/tex]
3. Apply Simpson's Rule formula:
[tex]\[ \int_1^5 \frac{6 \cos (4 x)}{x} \, dx \approx \frac{h}{3} \left[ f(x_0) + 4 \sum_{i=1, 3, 5, \cdots} f(x_i) + 2 \sum_{i=2, 4, 6, \cdots (n-2)} f(x_i) + f(x_n) \right] \][/tex]
4. The approximate value using Simpson's Rule to six decimal places is:
[tex]\[ 1.327371 \][/tex]
### Summary of results:
- Trapezoidal Rule: [tex]\(-1.901363\)[/tex]
- Midpoint Rule: [tex]\(2.941738\)[/tex]
- Simpson's Rule: [tex]\(1.327371\)[/tex]
