To evaluate the integral \(\int_{0}^{9} e^{2\sqrt{x}} \, dx\), we will perform a substitution to simplify the expression.

### Step-by-Step Solution

1. **Substitute**: Let \( u = 2\sqrt{x} \). Then, \( x = \frac{u^2}{4} \).

2. **Differentiate**: Compute \( dx \) in terms of \( du \).
   
   \[
   \frac{du}{dx} = \frac{d}{dx}(2\sqrt{x}) = \frac{2}{2\sqrt{x}} = \frac{1}{\sqrt{x}}
   \]

   Therefore, 

   \[
   dx = \frac{du}{2\sqrt{x}}
   \]

   Since \( \sqrt{x} = \frac{u}{2} \), we have:

   \[
   dx = \frac{du}{2 \cdot \frac{u}{2}} = \frac{du}{u}
   \]

   Hence,

   \[
   dx = \frac{du}{2u}
   \]

3. **Change of limits**: When \( x = 0 \), \( u = 2\sqrt{0} = 0 \). When \( x = 9 \), \( u = 2\sqrt{9} = 6 \).

4. **Rewrite the integral**: The integral in terms of \( u \) is:

   \[
   \int_{0}^{9} e^{2\sqrt{x}} \, dx = \int_{0}^{6} e^{u} \cdot \frac{u}{4} \, du
   \]

   Simplify:

   \[
   \int_{0}^{6} \frac{e^u}{u} \cdot \frac{u}{4} \, du = \frac{1}{4} \int_{0}^{6} ue^u \, du
   \]

5. **Integrate**: Integrate \( e^u \) with respect to \( u \).

   \[
   \int ue^u \, du = e^u(u - 1) + C
   \]

6. **Evaluate definite integral**: Apply the limits from 0 to 6.

   \[
   \left[\frac{1}{4} \left(e^u(u - 1)\right)\right]_{0}^{6} = \frac{1}{4} \left( (6 - 1)e^6 - (0 - 1)e^0 \right)
   \]

   \[
   = \frac{1}{4} \left(5e^6 + 1\right)
   \]

7. **Simplify**:

   \[
   = \frac{5e^6 + 1}{4}
   \]

### Final Answer

\[
\int_{0}^{9} e^{2\sqrt{x}} \, dx = \frac{5e^6 + 1}{4}
\]

Would you like more details or have any questions?

Here are 5 related questions for further exploration:
1. How would you solve \(\int_{0}^{9} e^{3\sqrt{x}} \, dx\)?
2. What happens if the limits of integration change, for example, from 1 to 4?
3. How does integration by parts work, and when is it useful?
4. Can you derive the formula for \(\int e^{ax} \, dx\) from scratch?
5. What is the importance of substitution in integration, and how does it work?

**Tip**: Always check the limits of integration when substituting variables to ensure they match the new variable's range.

Learn how to evaluate the integral of e^(2*sqrt(x)) from 0 to 9 using the substitution method. This detailed step-by-step solution covers integral calculus, substitution techniques, and exponential functions. Suitable for advanced college students and those interested in calculus.

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