To find the integral of \( e^{7x - 2} \, dx \), we can use a basic rule for integrating exponential functions. The function inside the integral is \( e^{7x - 2} \), which is an exponential function of the form \( e^{u} \), where \( u = 7x - 2 \).

### Steps to Solve:

1. **Substitute \( u \):**
   Let \( u = 7x - 2 \). 

2. **Differentiate \( u \):**
   \[
   \frac{du}{dx} = 7 \implies du = 7 \, dx \implies dx = \frac{du}{7}.
   \]

3. **Rewrite the integral:**
   Substitute into the integral:
   \[
   \int e^{7x - 2} \, dx = \int e^u \cdot \frac{du}{7} = \frac{1}{7} \int e^u \, du.
   \]

4. **Integrate with respect to \( u \):**
   The integral of \( e^u \) with respect to \( u \) is simply \( e^u \). So:
   \[
   \frac{1}{7} \int e^u \, du = \frac{1}{7} e^u + C.
   \]

5. **Substitute \( u = 7x - 2 \) back into the result:**
   \[
   \frac{1}{7} e^{7x - 2} + C.
   \]

### Final Answer:

\[
\int e^{7x - 2} \, dx = \frac{1}{7} e^{7x - 2} + C.
\]

Would you like further details, or do you have any questions? Here are some related questions:

1. How do you integrate more complex exponential functions?
2. What are some applications of exponential integrals in real-world problems?
3. How would this process change if the exponent were a more complex function of \(x\)?
4. How can substitution methods be applied to other types of integrals?
5. What techniques can simplify the integration of functions involving \(e^x\)?

**Tip:** Always double-check your substitutions, especially when dealing with exponential functions, as small errors can lead to incorrect integrals.

Discover how to integrate e^(7x - 2) step-by-step with detailed explanations and examples. This problem covers integration techniques, specifically for exponential functions using substitution. Suitable for advanced high school and college students.

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