The integral to solve is:

\[
\int_0^9 e^{2x} \, dx
\]

To solve this definite integral, we'll follow these steps:

### Step 1: Find the indefinite integral
First, compute the indefinite integral of \( e^{2x} \).

\[
\int e^{2x} \, dx
\]

Let \( u = 2x \), so \( du = 2 \, dx \), or \( \frac{du}{2} = dx \).

The integral becomes:

\[
\int e^{u} \cdot \frac{1}{2} \, du = \frac{1}{2} \int e^{u} \, du = \frac{1}{2} e^{u} + C
\]

Substituting back \( u = 2x \):

\[
\frac{1}{2} e^{2x} + C
\]

### Step 2: Evaluate the definite integral
Now, apply the limits from 0 to 9:

\[
\int_0^9 e^{2x} \, dx = \left[ \frac{1}{2} e^{2x} \right]_0^9
\]

Substitute the limits:

\[
\frac{1}{2} \left( e^{2 \times 9} - e^{2 \times 0} \right) = \frac{1}{2} \left( e^{18} - e^{0} \right)
\]

Since \( e^0 = 1 \), this simplifies to:

\[
\frac{1}{2} \left( e^{18} - 1 \right)
\]

So, the result of the integral is:

\[
\boxed{\frac{1}{2} \left( e^{18} - 1 \right)}
\]

Would you like further details on any step or additional explanation?

### Related Questions:
1. How do you find the indefinite integral of \( e^{kx} \) for any constant \( k \)?
2. What is the process of solving definite integrals with exponential functions?
3. How can substitution be used effectively in integrals?
4. Why is the constant of integration ignored in definite integrals?
5. How does changing the limits of integration affect the outcome?

**Tip:** Always double-check the substitution step, as errors there can lead to incorrect results!

Learn how to solve the definite integral ∫0^9 e^{2x} dx using step-by-step methods. This problem involves integration of exponential functions and applying definite integral limits. Suitable for advanced high school students and beyond.

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