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To evaluate the integral \( \int e^{4x - 2} \, dx \), we can solve it step by step:

1. First, observe that the integrand is an exponential function of the form \( e^{ax + b} \), where \( a = 4 \) and \( b = -2 \).

2. To integrate this, we'll use substitution. Let:
   \[
   u = 4x - 2
   \]
   Then, the derivative of \( u \) with respect to \( x \) is:
   \[
   \frac{du}{dx} = 4 \quad \text{or} \quad dx = \frac{du}{4}
   \]

3. Now, substitute into the integral:
   \[
   \int e^{4x - 2} \, dx = \int e^u \cdot \frac{du}{4}
   \]

4. Simplify the integral:
   \[
   \frac{1}{4} \int e^u \, du
   \]

5. The integral of \( e^u \) is just \( e^u \), so we get:
   \[
   \frac{1}{4} e^u + C
   \]

6. Finally, substitute \( u = 4x - 2 \) back into the expression:
   \[
   \frac{1}{4} e^{4x - 2} + C
   \]

Thus, the solution to the integral is:
\[
\int e^{4x - 2} \, dx = \frac{1}{4} e^{4x - 2} + C
\]

Would you like more details or have any further questions?

Here are 5 related questions:
1. How would you evaluate \( \int e^{5x + 1} \, dx \)?
2. What happens if the exponent in the integral \( e^{ax + b} \) has a different base, like \( a^x \)?
3. Can you solve the integral \( \int x e^{4x - 2} \, dx \)?
4. What is the general formula for integrating functions of the form \( e^{ax + b} \)?
5. How do definite integrals differ from indefinite integrals?

**Tip:** When using substitution, always remember to change both the variable and the differential.

Evaluate the integral \( \int e^{4x - 2} \, dx \).

Learn how to evaluate the integral \( \int e^{4x - 2} \, dx \) using substitution. This solution breaks down the steps for integrating exponential functions, highlighting key concepts such as substitution and the integral of \( e^{ax + b} \). Suitable for students in Grades 11-12.

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