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resuélvela en integración por partes de una integral 

The image you uploaded contains a set of integrals that are to be solved using integration by parts. Let's solve one of them step by step.

### I'll start with the first integral:

\[
\int x^2 e^{3x} \, dx
\]

**Step 1: Identify parts for integration by parts:**
- Choose \( u = x^2 \) and \( dv = e^{3x} dx \).

**Step 2: Differentiate and integrate:**
- Differentiate \( u \):
  \[
  du = 2x \, dx
  \]
- Integrate \( dv \):
  \[
  v = \frac{1}{3} e^{3x}
  \]

**Step 3: Apply the integration by parts formula:**

The formula for integration by parts is:
\[
\int u \, dv = uv - \int v \, du
\]
Substitute the identified parts:
\[
\int x^2 e^{3x} \, dx = x^2 \cdot \frac{1}{3} e^{3x} - \int \frac{1}{3} e^{3x} \cdot 2x \, dx
\]

Simplify the expression:
\[
= \frac{1}{3} x^2 e^{3x} - \frac{2}{3} \int x e^{3x} \, dx
\]

**Step 4: Apply integration by parts again for the remaining integral:**

Now we need to integrate \( \int x e^{3x} \, dx \). Use integration by parts again with:
- \( u = x \) and \( dv = e^{3x} dx \).

Differentiate and integrate:
- \( du = dx \)
- \( v = \frac{1}{3} e^{3x} \)

Apply integration by parts again:
\[
\int x e^{3x} \, dx = x \cdot \frac{1}{3} e^{3x} - \int \frac{1}{3} e^{3x} \cdot 1 \, dx
\]
\[
= \frac{1}{3} x e^{3x} - \frac{1}{3} \int e^{3x} \, dx
\]
\[
= \frac{1}{3} x e^{3x} - \frac{1}{9} e^{3x}
\]

**Step 5: Substitute back into the original equation:**

\[
\int x^2 e^{3x} \, dx = \frac{1}{3} x^2 e^{3x} - \frac{2}{3} \left( \frac{1}{3} x e^{3x} - \frac{1}{9} e^{3x} \right)
\]

Simplify the expression:
\[
= \frac{1}{3} x^2 e^{3x} - \frac{2}{9} x e^{3x} + \frac{2}{27} e^{3x} + C
\]

So the solution to the integral is:
\[
\int x^2 e^{3x} \, dx = \frac{1}{3} x^2 e^{3x} - \frac{2}{9} x e^{3x} + \frac{2}{27} e^{3x} + C
\]

Would you like to go through the solution for another integral or have any questions about this one?

### Relative questions:
1. How does integration by parts relate to the product rule in differentiation?
2. What are the key indicators that an integral can be solved using integration by parts?
3. Could you use a different choice for \( u \) and \( dv \) in the given integral? How would that affect the solution?
4. How would you handle an integral where both parts \( u \) and \( dv \) are functions that do not simplify easily?
5. What are some alternative methods to integration by parts for solving complex integrals?

**Tip:** Always look for simpler functions when choosing \( u \) in integration by parts; this simplifies the process and often reduces the number of steps.

Explore a detailed step-by-step solution to the integral of x^2 * e^(3x) using integration by parts. This problem involves applying the integration by parts formula and simplifying the resulting expression. Suitable for college-level students and those studying integration techniques.

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