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The integral provided is:

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

To integrate this expression, we'll first notice that it's more straightforward to perform a substitution rather than using integration by parts.

### Step 1: Substitution
Let's substitute:
\[
u = 3\sqrt{x}
\]
Then,
\[
\frac{du}{dx} = \frac{3}{2\sqrt{x}} \implies du = \frac{3}{2\sqrt{x}} \, dx
\]
Thus,
\[
dx = \frac{2\sqrt{x}}{3} \, du
\]

Since \( u = 3\sqrt{x} \), we have \( \sqrt{x} = \frac{u}{3} \). Substituting in \( dx \):

\[
dx = \frac{2\sqrt{x}}{3} \, du = \frac{2 \cdot \frac{u}{3}}{3} \, du = \frac{2u}{9} \, du
\]

Substitute this back into the integral:

\[
\int 3e^{3\sqrt{x}} \, dx = \int 3e^u \cdot \frac{2u}{9} \, du = \frac{6}{9} \int u e^u \, du = \frac{2}{3} \int u e^u \, du
\]

### Step 2: Integration by Parts
Now, apply integration by parts on \( \int u e^u \, du \). Let:
\[
u_1 = u, \quad dv = e^u \, du
\]
Then,
\[
du_1 = du, \quad v = e^u
\]

Apply the integration by parts formula:
\[
\int u e^u \, du = u e^u - \int e^u \, du
\]

Simplify the integral:
\[
\int u e^u \, du = u e^u - e^u + C_1 = e^u(u - 1) + C_1
\]

### Step 3: Substitute Back
Now, substituting back \( u = 3\sqrt{x} \):

\[
\frac{2}{3} \int u e^u \, du = \frac{2}{3} \left[ e^{3\sqrt{x}}(3\sqrt{x} - 1) \right] + C_2
\]

So the final answer is:

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

where \( C \) is the constant of integration.

Would you like more details on any part of this solution, or do you have any questions?

Here are some related questions:

1. How would you solve \(\int e^{\sqrt{x}} \, dx\)?
2. What substitution would you use for the integral \(\int 2x e^{2x} \, dx\)?
3. Can you integrate \(\int e^{ax + b} \, dx\) directly?
4. What is the process for reversing a substitution in an integral?
5. How does integration by parts compare to substitution for different types of integrals?

**Tip:** Always look for substitution opportunities first before applying integration by parts, especially when dealing with functions that involve square roots or other compositions.

Learn how to integrate the function 3e^(3sqrt(x)) using integration by parts and substitution. This detailed step-by-step solution covers key concepts like integration techniques, substitution method, and advanced mathematical methods for integration. Suitable for advanced mathematics students and enthusiasts.

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