To find the integral of $4x e^{2x} \, dx$, we can use integration by parts.

Step-by-Step Solution:

We'll apply the integration by parts formula:

$\int u \, dv = uv - \int v \, du$

Let:

• $u = 4x$, so $du = 4 \, dx$
• $dv = e^{2x} \, dx$, so $v = \frac{1}{2} e^{2x}$

Now, apply the integration by parts formula:

$\int 4x e^{2x} \, dx = 4x \cdot \frac{1}{2} e^{2x} - \int \frac{1}{2} e^{2x} \cdot 4 \, dx$

Simplifying:

$= 2x e^{2x} - \int 2 e^{2x} \, dx$

Next, integrate $2 e^{2x}$:

$\int 2 e^{2x} \, dx = \frac{2}{2} e^{2x} = e^{2x}$

Thus, the final result is:

$2x e^{2x} - e^{2x} + C$

Where $C$ is the constant of integration.

Final Answer:

$\int 4x e^{2x} \, dx = 2x e^{2x} - e^{2x} + C$

Would you like to explore the steps further or have any questions? Here are some related questions to expand on:

1. What is the integration by parts formula, and how can it be generalized?
2. How do you use integration by parts for functions with more than one term?
3. What happens if you apply a different substitution in this integral?
4. Can this approach be applied to other exponential functions involving polynomials?
5. How do you determine the constant of integration $C$ in real-world problems?

Tip: Always look for parts of the integrand that can be easily differentiated and integrated when using integration by parts.