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.