The given expression is:

$\frac{3}{297} \int_{3}^{300} \left( 4.41 \ln \left(\frac{300}{x}\right) \right) dx$

To solve this, we'll proceed with the integration step-by-step:

Step 1: Simplify the Expression

The constant $\frac{3}{297}$ can be simplified to $\frac{1}{99}$.

So, the expression simplifies to: $\frac{1}{99} \int_{3}^{300} 4.41 \ln \left(\frac{300}{x}\right) dx$

Step 2: Factor Out the Constant from the Integral

Since 4.41 is a constant, we can factor it out of the integral: $\frac{4.41}{99} \int_{3}^{300} \ln \left(\frac{300}{x}\right) dx$

This simplifies to: $\frac{4.41}{99} \int_{3}^{300} \ln \left(300\right) - \ln(x) \, dx$

Step 3: Integrate

The integral can now be split into two separate integrals: $\frac{4.41}{99} \left[ \int_{3}^{300} \ln(300) \, dx - \int_{3}^{300} \ln(x) \, dx \right]$

1. Integrating $\int_{3}^{300} \ln(300) \, dx$: Since $\ln(300)$ is a constant, the integral becomes: $\ln(300) \cdot (300 - 3)$ $\ln(300) \cdot 297$

2. Integrating $\int_{3}^{300} \ln(x) \, dx$: Use integration by parts where $u = \ln(x)$ and $dv = dx$.

   This gives us: $\left[ x \ln(x) \right]_{3}^{300} - \int_{3}^{300} x \cdot \frac{1}{x} \, dx$ Simplifying further: $\left[ x \ln(x) - x \right]_{3}^{300}$ Substituting the limits: [ \left[ 300 \ln(300) - 300 \right] - \left[ 3 \ln(3) - 3 \right] ] $300 \ln(300) - 300 - (3 \ln(3) - 3)$

Step 4: Combine and Simplify

Now, substituting the results back into the original expression:

$\frac{4.41}{99} \left[ 297 \ln(300) - (300 \ln(300) - 300 - 3 \ln(3) + 3) \right]$

Finally, simplify the expression to get the final result.

Would you like a detailed calculation of the numerical result, or do you have any other questions?

Here are five related questions to expand your understanding:

1. How does integration by parts work, and why is it useful in this context?
2. What is the significance of the natural logarithm function in this problem?
3. How do we apply the limits of integration after performing the integration?
4. Can the integral be evaluated using numerical methods if it is difficult to solve analytically?
5. What are some common applications of definite integrals in real-world problems?

Tip: When performing integration by parts, always carefully choose $u$ and $dv$ to simplify the problem as much as possible.