https://file.aiquickdraw.com/mathbot/file/1727094270471myqqfkya.jpg

The problem in the image asks to evaluate the integral:

\[
\int_e^{e^2} \frac{4}{x} \, dx
\]

### Solution:
The integral of \(\frac{4}{x}\) with respect to \(x\) is straightforward. We use the fact that:

\[
\int \frac{1}{x} \, dx = \ln|x|
\]

So, we calculate:

\[
\int \frac{4}{x} \, dx = 4 \ln |x|
\]

Now, applying the limits of integration from \(e\) to \(e^2\):

\[
\left[ 4 \ln |x| \right]_e^{e^2} = 4 \ln (e^2) - 4 \ln (e)
\]

Using the properties of logarithms:

\[
4 \ln (e^2) = 4 \cdot 2 \ln(e) = 8 \ln(e) = 8
\]

\[
4 \ln (e) = 4
\]

Thus, the result is:

\[
8 - 4 = 4
\]

### Final Answer:
The correct answer is \( \boxed{4} \).

---

Would you like further details on this solution or have any questions? Here are five related questions you might find interesting:

1. How do you evaluate a logarithmic integral with different limits?
2. What are the properties of logarithms that simplify integrals like these?
3. How do definite integrals work when limits involve exponential functions?
4. What role do the limits of integration play in calculating definite integrals?
5. How can you apply this method to similar integrals with more complex expressions?

**Tip:** When dealing with logarithmic integrals, always remember to apply the properties of logarithms to simplify your result after evaluating at the limits.

Find the integral ∫(e to e²) (4/x) dx.

Learn how to evaluate the definite integral ∫(e to e²) (4/x) dx using logarithmic properties and the Fundamental Theorem of Calculus. This problem explores key calculus concepts such as definite integrals and logarithmic functions. Suitable for undergraduate students studying calculus.

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