find the integral of (3^(3sqrtx-2))/2sqrtx dx

We are tasked with finding the integral of the function:

\[
\int \frac{3^{3\sqrt{x} - 2}}{2\sqrt{x}} \, dx
\]

### Step-by-Step Solution:

#### Step 1: Substitution
Let’s first set a substitution to simplify the expression:

Let \( u = \sqrt{x} = x^{1/2} \), so that:
\[
\frac{du}{dx} = \frac{1}{2\sqrt{x}} \implies dx = 2u \, du
\]

This transforms \( \sqrt{x} \) to \( u \), and the integral becomes:
\[
\int \frac{3^{3u - 2}}{2u} \cdot 2u \, du
\]

The \( 2u \) terms cancel out, and we are left with:
\[
\int 3^{3u - 2} \, du
\]

#### Step 2: Break the exponent
We can express the exponent \( 3u - 2 \) as:
\[
3^{3u - 2} = \frac{3^{3u}}{3^2} = \frac{3^{3u}}{9}
\]

Thus, the integral becomes:
\[
\frac{1}{9} \int 3^{3u} \, du
\]

#### Step 3: Another substitution
Now, let’s perform another substitution. Let \( v = 3u \), so that:
\[
dv = 3 \, du \implies du = \frac{dv}{3}
\]

Thus, the integral becomes:
\[
\frac{1}{9} \int 3^v \cdot \frac{dv}{3} = \frac{1}{27} \int 3^v \, dv
\]

#### Step 4: Integrate
The integral of \( 3^v \) with respect to \( v \) is:
\[
\int 3^v \, dv = \frac{3^v}{\ln(3)}
\]

Thus, the result of the integral is:
\[
\frac{1}{27} \cdot \frac{3^v}{\ln(3)} = \frac{3^v}{27\ln(3)}
\]

#### Step 5: Back-substitute \( v \) and \( u \)
Now, substitute back \( v = 3u \) and \( u = \sqrt{x} \):
\[
\frac{3^{3u}}{27\ln(3)} = \frac{3^{3\sqrt{x}}}{27\ln(3)}
\]

Thus, the final answer is:
\[
\int \frac{3^{3\sqrt{x} - 2}}{2\sqrt{x}} \, dx = \frac{3^{3\sqrt{x}}}{27\ln(3)} + C
\]

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

---

Would you like more details or have any questions?

Here are 5 related questions for further practice:
1. What is the integral of \( \frac{e^{\sqrt{x}}}{\sqrt{x}} \, dx \)?
2. Find the derivative of \( 3^{3\sqrt{x}} \).
3. How do you perform substitution for integrals involving square roots?
4. Solve \( \int \frac{a^{\sqrt{x}}}{\sqrt{x}} \, dx \), where \( a \) is a constant.
5. Evaluate \( \int 3^{2x} \, dx \).

**Tip:** For integrals involving exponential functions, consider substitutions that simplify the exponent before integrating.

Learn how to solve the integral of (3^(3sqrtx-2))/2sqrtx with a detailed step-by-step guide. This problem covers integral calculus concepts such as substitution methods and exponential function integration. Ideal for students in undergraduate Calculus I/II.

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