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The problem asks to calculate the definite integral:

\[
\int_0^\pi e^{2x} \, dx
\]

### Step-by-Step Solution:

1. **Identify the function to integrate**: The function is \( f(x) = e^{2x} \).

2. **Set up the integral**: 
   \[
   \int e^{2x} \, dx
   \]

3. **Find the indefinite integral**:
   - To integrate \( e^{2x} \), we apply the rule \( \int e^{ax} \, dx = \frac{1}{a} e^{ax} + C \), where \( a \) is a constant.
   - Here, \( a = 2 \), so:
     \[
     \int e^{2x} \, dx = \frac{1}{2} e^{2x} + C
     \]

4. **Evaluate the definite integral** from 0 to \( \pi \):
   \[
   \int_0^\pi e^{2x} \, dx = \left[ \frac{1}{2} e^{2x} \right]_0^\pi
   \]
   Now substitute the limits:
   \[
   = \frac{1}{2} \left( e^{2\pi} - e^0 \right)
   \]
   Since \( e^0 = 1 \), this simplifies to:
   \[
   = \frac{1}{2} \left( e^{2\pi} - 1 \right)
   \]

### Final Answer:
\[
\int_0^\pi e^{2x} \, dx = \frac{1}{2} \left( e^{2\pi} - 1 \right)
\]

Would you like more details on any part of this solution? 

Here are some related questions to explore further:
1. How do you compute more complex exponential integrals?
2. What are the steps to integrate trigonometric functions?
3. Can definite integrals always be computed using the Fundamental Theorem of Calculus?
4. How do limits of integration affect the result of a definite integral?
5. How can you apply substitution to more difficult integrals?

**Tip**: When integrating exponential functions, always check if the exponent can be simplified through substitution to make integration easier.

Beräkna den bestämda integralen ∫₀^π e²ˣ dx.

Learn how to solve the definite integral ∫₀^π e²ˣ dx with a step-by-step solution. This problem explores calculus concepts such as integration of exponential functions and applying the Fundamental Theorem of Calculus. Ideal for college-level calculus students.

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