Final answer:
To approximate the integral of 5 √ ln(x) using the Trapezoidal Rule, Midpoint Rule, and Simpson's Rule with n=6, we need to find the width of the subintervals, evaluate the function as per each rule's formula, and round the results to six decimal places.
Step-by-step explanation:
To approximate the integral of 5 √ ln(x) from 1 to 4 using n=6 with the Trapezoidal Rule, Midpoint Rule, and Simpson's Rule, we'll need to calculate the width of each subinterval, which is (b - a) / n, where (b - a) is the interval from 1 to 4. In this case, the width is (4 - 1) / 6 = 0.5.
We will then evaluate the function at the necessary points and sum up the results according to the formulas specific to each rule to get the approximations.
With the Trapezoidal Rule, we add the function values at the endpoints, twice the function values at each interior point, and then multiply by the width of the subintervals divided by 2. For the Midpoint Rule, we evaluate the function at the midpoint of each subinterval and then multiply each value by the width of the subintervals.
Simpson's Rule is a bit more complex, we'll need to use the function values at the endpoints, four times the function values at the midpoints of the subintervals, and twice the function values at the other points, then multiply the sum by the width of the subintervals divided by 3.
Finally, we will round each approximation to six decimal places as required.