https://file.aiquickdraw.com/mathbot/file/172886829876538590651.jpg

The integral presented in the image is:

\[
\int \frac{21 e^{\sqrt{3x}}}{2 \sqrt{x}} \, dx
\]

To evaluate this, let's proceed step by step:

### Step 1: Simplifying the integral
We notice that the integral contains an exponential with \(\sqrt{3x}\) in the exponent and a square root in the denominator. A useful substitution for this type of problem is:

Let:
\[
u = \sqrt{3x}
\]

Then:
\[
u = \sqrt{3} \cdot \sqrt{x} \quad \Rightarrow \quad du = \frac{\sqrt{3}}{2 \sqrt{x}} \, dx
\]

We can also express \(dx\) in terms of \(du\):
\[
dx = \frac{2 \sqrt{x}}{\sqrt{3}} \, du
\]

### Step 2: Substituting into the integral
Now, substitute these into the original integral:
\[
\int \frac{21 e^{u}}{2 \sqrt{x}} \cdot \frac{2 \sqrt{x}}{\sqrt{3}} \, du
\]

Simplifying the expression:
\[
\int \frac{21}{\sqrt{3}} e^u \, du
\]

This simplifies to:
\[
7 \sqrt{3} \int e^u \, du
\]

### Step 3: Integrating
The integral of \(e^u\) is simply \(e^u\). Therefore, we get:
\[
7 \sqrt{3} e^u + C
\]

### Step 4: Substitute back \(u = \sqrt{3x}\)
Finally, substitute back \(u = \sqrt{3x}\) to get the final answer:
\[
7 \sqrt{3} e^{\sqrt{3x}} + C
\]

Thus, the correct answer is:

\[
\boxed{B} \ 7 \sqrt{3} e^{\sqrt{3x}} + C
\]

Would you like more details or have any questions?

### Follow-up questions:
1. What is the process of substitution in integrals?
2. How can you identify when to use substitution in an integral?
3. What are the rules for integrating exponential functions?
4. How does the constant \(C\) impact indefinite integrals?
5. Can you solve a similar integral with a different variable substitution?

### Tip:
When dealing with integrals involving exponential functions and square roots, substitution is often a useful technique to simplify the expression before integrating.

Evaluate the integral ∫ (21e^(√3x) / (2√x)) dx

Learn how to solve the integral of 21e^(√3x) / (2√x) step by step using substitution. This problem involves key concepts of exponential functions and integration techniques. Suitable for advanced high school students.

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