Use the Trapezoidal Rule to approximate ¡Ò8 4 ln(x2+5) dx using n=3. Round your answer to the fourth decimal place.

There are any number of calculators online to help you confirm your solution. One is here:

http://www.emathhelp.net/calculators/calculus-2/trapezoidal-rule-calculator/

To approximate the integral using the Trapezoidal Rule with n=3, follow these steps:

1. Determine the limits of integration. In this case, the limits are 8 and 4.

2. Calculate the width of each subinterval (Δx) by dividing the difference between the limits by the number of subintervals (n):
Δx = (b - a) / n,
where a is the lower limit (4), b is the upper limit (8), and n is the number of subintervals (3).

Δx = (8 - 4) / 3 = 4 / 3 = 1.3333 (rounded to four decimal places).

3. Set up the formula for the Trapezoidal Rule:
T ≈ [f(a) + 2 * f(x1) + 2 * f(x2) + ... + f(b)] * Δx / 2,
where f(x) is the function we are integrating, a is the lower limit, b is the upper limit, x1, x2, etc., are the sample points inside each subinterval, and Δx is the width of each subinterval.

4. Calculate the values of the function at the sample points. In this case, the function is ln(x^2 + 5), so we need to determine the values at x=a, x=x1, x=x2, and x=b.

For x=a (4):
f(4) = ln((4^2) + 5) = ln(16 + 5) = ln(21) ≈ 3.0445 (rounded to four decimal places).

For x=x1 (4 + Δx = 4 + 1.3333 = 5.3333):
f(5.3333) = ln((5.3333^2) + 5) = ln(28.4434) ≈ 3.3464 (rounded to four decimal places).

For x=x2 (5.3333 + Δx = 5.3333 + 1.3333 = 6.6666):
f(6.6666) = ln((6.6666^2) + 5) = ln(50.3336) ≈ 3.9188 (rounded to four decimal places).

For x=b (8):
f(8) = ln((8^2) + 5) = ln(69) ≈ 4.2341 (rounded to four decimal places).

5. Plug the values into the Trapezoidal Rule formula and calculate the approximation:
T ≈ [f(a) + 2 * f(x1) + 2 * f(x2) + ... + f(b)] * Δx / 2
≈ [3.0445 + 2 * 3.3464 + 2 * 3.9188 + 4.2341] * 1.3333 / 2
≈ [3.0445 + 6.6928 + 7.8376 + 4.2341] * 0.6666
≈ 21.669 (rounded to four decimal places).

Therefore, using the Trapezoidal Rule with n=3, the approximate value of the integral ∫8 to 4 of ln(x^2+5) dx is approximately 21.669.