We are tasked with approximating the integral:

$\int_1^5 \frac{7 \cos(x)}{x} \, dx$

using three numerical methods:

1. Trapezoidal Rule
2. Midpoint Rule
3. Simpson's Rule

The number of subintervals is $n = 8$, and we are asked to provide each answer correct to six decimal places.

Step-by-step solution outline:

Step 1: Define the step size $h$

For all three methods, the step size $h$ is given by:

$h = \frac{b - a}{n} = \frac{5 - 1}{8} = 0.5$

Step 2: Partition the interval

The points for the partition of the interval [1, 5] with step size $h = 0.5$ are:

$x_0 = 1, \quad x_1 = 1.5, \quad x_2 = 2, \quad x_3 = 2.5, \quad x_4 = 3, \quad x_5 = 3.5, \quad x_6 = 4, \quad x_7 = 4.5, \quad x_8 = 5$

(a) Trapezoidal Rule

The Trapezoidal Rule formula is:

$T_n = \frac{h}{2} \left[ f(x_0) + 2\sum_{i=1}^{n-1} f(x_i) + f(x_n) \right]$

(b) Midpoint Rule

The Midpoint Rule formula is:

$M_n = h \sum_{i=1}^{n} f\left( \frac{x_{i-1} + x_i}{2} \right)$

(c) Simpson's Rule

Simpson's Rule formula (for even $n$) is:

$S_n = \frac{h}{3} \left[ f(x_0) + 4 \sum_{\text{odd } i} f(x_i) + 2 \sum_{\text{even } i} f(x_i) + f(x_n) \right]$

Now, I will compute the values of the function $f(x) = \frac{7 \cos(x)}{x}$ at the relevant points for each rule and calculate the approximations to six decimal places.

I'll go ahead and process the calculations.Here are the approximate values for the integral $\int_1^5 \frac{7 \cos(x)}{x} \, dx$ using the three numerical methods with $n = 8$ subintervals, correct to six decimal places:

1. Trapezoidal Rule: $T_n = -3.467332$
2. Midpoint Rule: $M_n = -3.803249$
3. Simpson's Rule: $S_n = -3.682859$

Would you like further details or explanations on any of these methods?

Here are some follow-up questions you could explore:

1. How does increasing the number of subintervals $n$ affect the accuracy of each method?
2. Why is Simpson's Rule generally more accurate than the Trapezoidal and Midpoint Rules for smooth functions?
3. What happens to the accuracy if we apply these methods to an integral with a singularity or discontinuity?
4. Can we apply these numerical methods to multidimensional integrals? How does the approach differ?
5. How do these methods relate to the actual value of the integral? Could we find an exact solution?

Tip: Simpson's Rule is typically more accurate than the Trapezoidal and Midpoint Rules for functions that are smooth because it uses a quadratic approximation instead of linear.