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The given problem asks for the evaluation of the iterated integral:

\[
\int_0^{\ln 5} \int_1^{\ln 2} e^{2x + 3y} \, dy \, dx.
\]

Let’s solve this step by step.

---

### Step 1: Separate the exponential term
The expression \(e^{2x + 3y}\) can be written as the product of two independent terms:
\[
e^{2x + 3y} = e^{2x} \cdot e^{3y}.
\]

This allows us to separate the variables when integrating.

---

### Step 2: Inner integral (with respect to \(y\))
We evaluate:
\[
\int_1^{\ln 2} e^{3y} \, dy.
\]

The integral of \(e^{3y}\) is:
\[
\frac{e^{3y}}{3}.
\]

Apply the limits \(y = 1\) to \(y = \ln 2\):
\[
\left[ \frac{e^{3y}}{3} \right]_1^{\ln 2} = \frac{e^{3 \ln 2}}{3} - \frac{e^{3 \cdot 1}}{3}.
\]

Since \(e^{3 \ln 2} = (e^{\ln 2})^3 = 2^3 = 8\), this becomes:
\[
\frac{8}{3} - \frac{e^3}{3} = \frac{8 - e^3}{3}.
\]

---

### Step 3: Outer integral (with respect to \(x\))
The outer integral now becomes:
\[
\int_0^{\ln 5} e^{2x} \cdot \frac{8 - e^3}{3} \, dx.
\]

Since \(\frac{8 - e^3}{3}\) is a constant with respect to \(x\), we factor it out:
\[
\frac{8 - e^3}{3} \int_0^{\ln 5} e^{2x} \, dx.
\]

---

### Step 4: Solve the remaining integral
The integral of \(e^{2x}\) is:
\[
\frac{e^{2x}}{2}.
\]

Apply the limits \(x = 0\) to \(x = \ln 5\):
\[
\left[ \frac{e^{2x}}{2} \right]_0^{\ln 5} = \frac{e^{2 \ln 5}}{2} - \frac{e^{2 \cdot 0}}{2}.
\]

Since \(e^{2 \ln 5} = (e^{\ln 5})^2 = 5^2 = 25\) and \(e^0 = 1\), this becomes:
\[
\frac{25}{2} - \frac{1}{2} = \frac{24}{2} = 12.
\]

---

### Step 5: Combine results
The final result is:
\[
\frac{8 - e^3}{3} \cdot 12 = 4(8 - e^3).
\]

---

### Final Answer:
The correct choice is:
\[
\textbf{a. } 4(8 - e^3).
\]

Let me know if you'd like a more detailed explanation!

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### Related Questions:
1. How does separation of exponential terms work in integrals?
2. Can this problem be solved with reversed integration limits?
3. Why does the constant factor \(\frac{8 - e^3}{3}\) simplify the problem?
4. What is the significance of using iterated integrals instead of double integrals?
5. How would the answer change if \(e^{2x + 3y}\) was replaced by \(e^{2x - 3y}\)?

---

### Tip:
Always check whether you can separate variables in an exponential function—it often simplifies multivariable integrals significantly!

Evaluate the iterated integral \( \int_0^{\ln 5} \int_1^{\ln 2} e^{2x + 3y} \, dy \, dx \) and select the correct answer from the options.

Step-by-step solution for evaluating the iterated integral \( \int_0^{\ln 5} \int_1^{\ln 2} e^{2x + 3y} \, dy \, dx \). Covers concepts in multivariable calculus and exponential functions. Learn about Fubini's theorem and the integral of exponential functions.

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