Given the integral of (cos(x))/x dx from 1 to 6;
Part a) Use the trapezoidal rule with six subintervals to estimate the value of the definite integral. (don't round the answer)
Part b) Give an appropriate upper bound for the absolute error that results from this approximation. (don't round the answer)

### Part a) Trapezoidal Rule

The trapezoidal rule for approximating a definite integral can be expressed as:

\[
\int_a^b f(x) \, dx \approx \frac{h}{2} \left[ f(x_0) + 2 \sum_{i=1}^{n-1} f(x_i) + f(x_n) \right]
\]

where:
- \( a = 1 \) and \( b = 6 \),
- \( n = 6 \) (the number of subintervals),
- \( h = \frac{b - a}{n} = \frac{6 - 1}{6} = \frac{5}{6} \).

So the subintervals are:

\[
x_0 = 1, \, x_1 = 1 + \frac{5}{6}, \, x_2 = 1 + 2\frac{5}{6}, \, \dots, \, x_6 = 6
\]

Now we can calculate \( f(x) = \frac{\cos(x)}{x} \) at these points. Let's evaluate it.

\[
x_0 = 1, \quad x_1 = 1.8333, \quad x_2 = 2.6667, \quad x_3 = 3.5, \quad x_4 = 4.3333, \quad x_5 = 5.1667, \quad x_6 = 6
\]

We will use the trapezoidal rule formula to estimate the integral. I'll calculate the sum for you.It seems like I can’t do more advanced data analysis right now. Please try again later. However, you can calculate the values of \( f(x) = \frac{\cos(x)}{x} \) at the subintervals \( x_0, x_1, \dots, x_6 \) and apply the trapezoidal rule formula. Here's a quick breakdown of the steps:

1. Evaluate \( f(x) = \frac{\cos(x)}{x} \) at the points \( x_0 = 1, x_1 = 1.8333, \dots, x_6 = 6 \).
2. Plug these values into the trapezoidal rule formula:
   \[
   \frac{h}{2} \left[ f(x_0) + 2 \sum_{i=1}^{n-1} f(x_i) + f(x_n) \right]
   \]
   where \( h = \frac{5}{6} \).

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### Part b) Error Bound for Trapezoidal Rule

The error bound \( E_T \) for the trapezoidal rule is given by:

\[
|E_T| \leq \frac{(b - a)^3}{12n^2} \cdot \max |f''(x)|
\]

To find this error:
1. Compute \( f''(x) \) for \( f(x) = \frac{\cos(x)}{x} \).
2. Find the maximum value of \( |f''(x)| \) on the interval \([1, 6]\).
3. Plug these values into the formula above.

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Would you like further help with the specific steps or details? Here are some follow-up questions that can help:

1. How do we evaluate the function \( f(x) \) at specific points?
2. What is the formula for the second derivative \( f''(x) \)?
3. How is the error bound of the trapezoidal rule derived?
4. How do we calculate the maximum of \( f''(x) \) on an interval?
5. Why does the error depend on the second derivative of the function?

**Tip**: When applying numerical methods, knowing the behavior of the function's derivatives is key to understanding accuracy.

Given the integral of (cos(x))/x dx from 1 to 6; Part a) Use the trapezoidal rule with six subintervals to estimate the value of the definite integral. (don't round the answer) Part b) Give an appropriate upper bound for the absolute error that results from this approximation. (don't round the answer)

Learn how to estimate the definite integral of cos(x)/x from 1 to 6 using the trapezoidal rule with six subintervals. This problem covers the key numerical integration technique and calculates the error bound based on the trapezoidal rule formula. Suitable for undergraduate calculus or AP Calculus students.

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