To evaluate the integral \(\int_{0}^{5} e^{3x} \, dx\), we follow the steps of finding the antiderivative and then applying the limits of integration. 

1. **Find the antiderivative**:
   The antiderivative of \( e^{3x} \) is:
   \[
   \int e^{3x} \, dx = \frac{1}{3} e^{3x} + C
   \]

2. **Apply the limits of integration**:
   We now evaluate this antiderivative from 0 to 5:
   \[
   \int_{0}^{5} e^{3x} \, dx = \left[ \frac{1}{3} e^{3x} \right]_{0}^{5}
   \]

   This means we substitute \( x = 5 \) and \( x = 0 \) into the antiderivative and subtract:
   \[
   \left. \frac{1}{3} e^{3x} \right|_{0}^{5} = \frac{1}{3} e^{15} - \frac{1}{3} e^{0}
   \]

3. **Simplify the expression**:
   Since \( e^{0} = 1 \):
   \[
   \frac{1}{3} e^{15} - \frac{1}{3} \cdot 1 = \frac{1}{3} e^{15} - \frac{1}{3}
   \]

Therefore, the value of the definite integral is:
\[
\int_{0}^{5} e^{3x} \, dx = \frac{1}{3} (e^{15} - 1)
\]

Would you like any further details or explanations?

Here are some related questions you might find interesting:
1. How would you integrate \( e^{3x} \) if the limits were from 1 to 4?
2. What is the general formula for integrating \( e^{ax} \)?
3. How does the integration process change if the exponent is a quadratic function, such as \( e^{x^2} \)?
4. Can you solve the integral \( \int_{0}^{1} e^{3x^2} \, dx \)?
5. What are some common applications of integrals involving exponential functions in real life?
6. How would you integrate \( e^{3x + 2} \)?
7. What techniques can be used to integrate \( e^{3x} \sin(2x) \)?
8. How does substitution help in solving integrals of exponential functions?

**Tip:** When evaluating definite integrals, always remember to apply the limits of integration after finding the indefinite integral to avoid mistakes.

Learn how to evaluate the integral ∫(0 to 5) e^(3x) dx with a detailed step-by-step solution. This problem involves integrating exponential functions and applying the limits of integration. Suitable for advanced high school or college students.

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