To solve the integral:

$\int 4x \cdot e^{x} \, dx$

We can use integration by parts, which is based on the formula:

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

Step 1: Choose $u$ and $dv$

Let:

• $u = 4x$ (because its derivative simplifies),
• $dv = e^{x} \, dx$.

Step 2: Compute $du$ and $v$

• $du = 4 \, dx$,
• $v = \int e^{x} \, dx = e^{x}$.

Step 3: Apply the formula

Now, apply the integration by parts formula:

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

Step 4: Solve the remaining integral

The integral $\int 4 \cdot e^{x} \, dx$ is straightforward:

$\int 4 \cdot e^{x} \, dx = 4e^{x}$

Step 5: Combine the results

Now, substitute everything back:

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

Where $C$ is the constant of integration.

Final Answer:

$\int 4x \cdot e^{x} \, dx = 4e^{x} (x - 1) + C$

Would you like more details on integration by parts or related topics? Here's some additional questions for exploration:

1. How can integration by parts be generalized for multiple functions?
2. What is the integral of $x^2 \cdot e^x$?
3. How do you apply the tabular method to integration by parts?
4. What is the role of the constant $C$ in indefinite integrals?
5. How does the integration by parts technique relate to the product rule in differentiation?

Tip: Always check if simplifying the integrand before applying integration by parts can make the process easier.