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The problem asks to find the indefinite integral of the expression and check the result by differentiation:

\[
\int \left( x^{3/2} + 6x + 9 \right) dx
\]

### Step-by-step Solution

#### 1. Integrating each term separately:

- The integral of \( x^{3/2} \):
\[
\int x^{3/2} dx = \frac{2}{5} x^{5/2}
\]
Here we use the power rule: \( \int x^n dx = \frac{x^{n+1}}{n+1} \).

- The integral of \( 6x \):
\[
\int 6x dx = 3x^2
\]
Since \( \int x dx = \frac{x^2}{2} \), multiplying by 6 gives \( 3x^2 \).

- The integral of the constant 9:
\[
\int 9 dx = 9x
\]

#### 2. Adding them together:
\[
\int \left( x^{3/2} + 6x + 9 \right) dx = \frac{2}{5} x^{5/2} + 3x^2 + 9x + C
\]
where \( C \) is the constant of integration.

### 3. Checking by differentiation:
Now, let's differentiate the result to verify:

\[
\frac{d}{dx} \left( \frac{2}{5} x^{5/2} + 3x^2 + 9x + C \right)
\]

- The derivative of \( \frac{2}{5} x^{5/2} \) is \( x^{3/2} \).
- The derivative of \( 3x^2 \) is \( 6x \).
- The derivative of \( 9x \) is \( 9 \).
- The derivative of the constant \( C \) is 0.

So:
\[
\frac{d}{dx} \left( \frac{2}{5} x^{5/2} + 3x^2 + 9x + C \right) = x^{3/2} + 6x + 9
\]
This matches the original integrand, confirming the solution is correct.

### Final Answer:
\[
\frac{2}{5} x^{5/2} + 3x^2 + 9x + C
\]

Would you like more details on any step? Here are some related questions:

1. How do you use the power rule of integration for different exponents?
2. Can you explain why a constant like \( C \) is added in indefinite integrals?
3. How would the process change for definite integrals?
4. What is the difference between indefinite and definite integrals?
5. How is the constant of integration \( C \) determined in applications?

**Tip:** Always check your integral by differentiating the result to ensure accuracy.

Find the indefinite integral and check the result by differentiation: ∫(x^(3/2) + 6x + 9) dx

Learn how to compute the indefinite integral of x^(3/2) + 6x + 9 and verify the result using differentiation. This problem involves applying the power rule of integration and understanding the relationship between integration and differentiation.

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