To solve the integral $\int 4x e^{4x} \, dx$, we will use integration by parts. The formula for integration by parts is:

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

Step 1: Choose $u$ and $dv$

Let:

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

Step 2: Apply the formula

$\int 4x e^{4x} \, dx = u v - \int v \, du$ Substituting $u$, $v$, and $du$: $\int 4x e^{4x} \, dx = \left(4x \cdot \frac{1}{4} e^{4x}\right) - \int \left(\frac{1}{4} e^{4x} \cdot 4\right) \, dx$

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

Step 3: Solve the remaining integral

$\int e^{4x} \, dx = \frac{1}{4} e^{4x}$.

Substitute this back: $\int 4x e^{4x} \, dx = x e^{4x} - \frac{1}{4} e^{4x} + C$

Final Answer:

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

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Let me know if you'd like further clarification!

Related Questions:

1. How does integration by parts work, and when should it be used?
2. What are common functions for which integration by parts is useful?
3. Can you verify the solution by differentiation?
4. How do you integrate $x^2 e^{4x}$?
5. What happens if the function has a different exponential, like $e^{2x}$?

Tip:

Always verify your integration by differentiating the result to ensure it matches the original integrand.