Certainly! Let's tackle this problem step-by-step.
### Problem Statement
We are given the following limit:
[tex]\[\lim_{n \rightarrow \infty} \sum_{i=1}^n \frac{\cos \left(x_i\right)}{x_i} \Delta x, \text{ on the interval } [3\pi, 4\pi]\][/tex]
This expression is the definition of a Riemann sum, which approximates the integral of a function over a given interval. As [tex]\(n\)[/tex] approaches infinity, the Riemann sum converges to the definite integral of the function over that interval.
### Step-by-Step Solution
1. Identify the function and interval:
We need to identify the function [tex]\( f(x) \)[/tex] that is being summed. Here, the function is:
[tex]\[ f(x) = \frac{\cos(x)}{x} \][/tex]
The interval over which we are integrating is:
[tex]\[ [3\pi, 4\pi] \][/tex]
2. Express the limit as a definite integral:
The limit provided is:
[tex]\[ \lim_{n \rightarrow \infty} \sum_{i=1}^n \frac{\cos(x_i)}{x_i} \Delta x \][/tex]
This, by definition, is the Riemann sum for the integral of [tex]\( f(x) \)[/tex] over [tex]\([3\pi, 4\pi]\)[/tex]. Thus, we can express this limit as the following definite integral:
[tex]\[ \int_{3\pi}^{4\pi} \frac{\cos(x)}{x} \, dx \][/tex]
3. Set up the integral:
We now write down the required integral:
[tex]\[ \int_{3\pi}^{4\pi} \frac{\cos(x)}{x} \, dx \][/tex]
4. Evaluate the integral:
The integral:
[tex]\[ \int_{3\pi}^{4\pi} \frac{\cos(x)}{x} \, dx \][/tex]
is not one of the standard integrals that can be computed easily with basic techniques such as substitution or integration by parts. However, it can be evaluated using more advanced techniques or special functions.
Result:
The result of evaluating the integral:
[tex]\[ \int_{3\pi}^{4\pi} \frac{\cos(x)}{x} \, dx \][/tex]
is given by:
[tex]\[ -\text{Ci}(3\pi) + \text{Ci}(4\pi) \][/tex]
Here, [tex]\(\text{Ci}(x)\)[/tex] denotes the cosine integral function, which is a special function related to the integral of cosine divided by the variable.
5. Final Answer:
Hence, the final answer to the problem is:
[tex]\[ \int_{3\pi}^{4\pi} \frac{\cos(x)}{x} \, dx = -\text{Ci}(3\pi) + \text{Ci}(4\pi) \][/tex]
This expresses the limit as a definite integral and provides the evaluated result using the cosine integral function.
