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The problem shown in the image asks how the given integral changes after making the substitution \( e^{2x} = t \).

The original integral is:

\[
\int_{-\frac{1}{2} \ln 3}^{0} \frac{dx}{3 + e^{2x}}
\]

### Step-by-Step Solution:

1. **Substitution:**
   \[
   e^{2x} = t
   \]
   Differentiating this with respect to \(x\):
   \[
   2e^{2x}dx = dt \quad \text{or} \quad dx = \frac{dt}{2t}
   \]

2. **Changing the limits of integration:**

   For \(x = 0\):
   \[
   e^{2(0)} = 1 \quad \Rightarrow \quad t = 1
   \]

   For \(x = -\frac{1}{2} \ln 3\):
   \[
   e^{2 \left(-\frac{1}{2} \ln 3\right)} = e^{-\ln 3} = \frac{1}{3} \quad \Rightarrow \quad t = \frac{1}{3}
   \]

3. **Substituting into the integral:**

   The integral becomes:
   \[
   \int_{\frac{1}{3}}^1 \frac{\frac{dt}{2t}}{3 + t}
   \]
   This simplifies to:
   \[
   \frac{1}{2} \int_{\frac{1}{3}}^1 \frac{dt}{t(3 + t)}
   \]

Thus, the integral after substitution takes the form:
\[
\frac{1}{2} \int_{\frac{1}{3}}^1 \frac{dt}{t(3+t)}
\]

This matches the third option in the list.

### Answer:
The correct option is:
\[
\frac{1}{2} \int_{\frac{1}{3}}^1 \frac{dt}{t(3+t)}
\]

Let me know if you would like more details, and here are some related questions for further practice:

1. How do you compute the integral \( \int \frac{dx}{e^x + 1} \)?
2. What happens when you substitute \( e^x = t \) in an integral with a different exponent?
3. How do you handle integrals with logarithmic limits like \( \int_{-\ln a}^{\ln b} f(x) dx \)?
4. What are common techniques for solving integrals involving exponential functions?
5. How do you find the derivative of functions involving natural logarithms?

**Tip:** Always remember to adjust the limits of integration when making substitutions!

После подстановки e^{2x} = t, какой вид принимает интеграл \( \int_{-\frac{1}{2} \ln 3}^{0} \frac{dx}{3 + e^{2x}} \)?

Learn how to transform and solve the integral \( \int_{-\frac{1}{2} \ln 3}^{0} \frac{dx}{3 + e^{2x}} \) using the substitution method e^{2x} = t. This problem demonstrates the process of changing the variable and adjusting the limits of integration, with detailed step-by-step explanations suitable for undergraduate mathematics students.

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