Use the trapezoidal rule with n = 5 to approximate. Keep at least 2 decimal places accuracy.
Integrate: (cos(x))/(x) from x=1 to 5
To use the trapezoidal rule to approximate the integral of a function, we need to follow these steps:
1. Determine the number of subintervals, n. In this case, n = 5.
2. Calculate the width of each subinterval, h, by dividing the difference between the upper and lower bounds of integration by the number of subintervals: h = (5 - 1) / 5 = 4 / 5 = 0.8.
3. Create a table with n + 1 values of the function evaluated at equally spaced points within the interval. In this case, we need to evaluate the function at six points: x = 1, 1.8, 2.6, 3.4, 4.2, and 5.
4. Calculate the sum of the function values at the endpoints of the interval and twice the sum of the function values at the interior points. Let's call this sum S.
S = f(x0) + 2 * (f(x1) + f(x2) + ... + f(x4)) + f(x5)
5. Multiply S by h/2 to get the approximation of the integral.
Approximation = (h/2) * S
Now, let's calculate the approximation for the given integral using the trapezoidal rule with n = 5:
Step 1: n = 5
Step 2: h = (5 - 1) / 5 = 0.8
Step 3: Evaluate the function at the six points:
f(1) = cos(1) / 1 = 0.5403023058681398
f(1.8) = cos(1.8) / 1.8 = 0.4917804188301559
f(2.6) = cos(2.6) / 2.6 = 0.28251214208777466
f(3.4) = cos(3.4) / 3.4 = 0.13419281984023508
f(4.2) = cos(4.2) / 4.2 = -0.017313116086634827
f(5) = cos(5) / 5 = -0.1917848549326277
Step 4: Calculate the sum S:
S = 0.5403023058681398 + 2 * (0.4917804188301559 + 0.28251214208777466 + 0.13419281984023508 + -0.017313116086634827) + -0.1917848549326277
= 0.5403023058681398 + 2 * 0.8913722646717317 + -0.1917848549326277
= 2.22004560138186
Step 5: Calculate the approximation:
Approximation = (0.8/2) * 2.22004560138186
≈ 0.888018240552744
Therefore, using the trapezoidal rule with n = 5, the approximation of the integral of (cos(x))/(x) from x = 1 to 5 is approximately 0.8880 (rounded to 4 decimal places).