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To evaluate the definite integral 

\[
\int_{0}^{5} e^{3x} \, dx
\]

we follow these steps:

1. **Find the indefinite integral**:
   The antiderivative of \( e^{3x} \) with respect to \( x \) can be found using the following rule:
   
   \[
   \int e^{ax} \, dx = \frac{1}{a} e^{ax} + C
   \]
   
   Here, \( a = 3 \). Therefore,

   \[
   \int e^{3x} \, dx = \frac{1}{3} e^{3x} + C
   \]

2. **Evaluate the definite integral**:
   We now evaluate the antiderivative at the upper limit and subtract the value of the antiderivative at the lower limit:
   
   \[
   \int_{0}^{5} e^{3x} \, dx = \left[ \frac{1}{3} e^{3x} \right]_{0}^{5}
   \]
   
   Substituting the limits of integration:
   
   \[
   \left. \frac{1}{3} e^{3x} \right|_{0}^{5} = \frac{1}{3} e^{15} - \frac{1}{3} e^{0}
   \]
   
   Since \( e^{0} = 1 \):
   
   \[
   \frac{1}{3} e^{15} - \frac{1}{3} \cdot 1 = \frac{1}{3} e^{15} - \frac{1}{3}
   \]
   
   Simplifying, we get:
   
   \[
   \frac{1}{3} (e^{15} - 1)
   \]

So, 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 definite integral ∫_{0}^{5} e^{3x} dx step-by-step. This problem involves finding the antiderivative of an exponential function, applying limits of integration, and simplifying the result. Suitable for advanced high school students and college-level learners.

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