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.