solve `int(x^(2)+6/x+9)dx`

Answer

To solve the integral \( \int \left( x^2 + \frac{6}{x} + 9 \right) dx \), we can break it down into simpler parts. Here’s the step-by-step solution:

### Step 1: Rewrite the Integral
We start with the integral:
\[ \int \left( x^2 + \frac{6}{x} + 9 \right) dx \]

### Step 2: Separate the Integral
We can separate the integral into three distinct parts:
\[ \int x^2 \, dx + \int \frac{6}{x} \, dx + \int 9 \, dx \]

### Step 3: Integrate Each Part
Now we will integrate each part separately.

1. **Integrate \( x^2 \)**:
\[ \int x^2 \, dx = \frac{x^{2+1}}{2+1} = \frac{x^3}{3} \]

2. **Integrate \( \frac{6}{x} \)**:
\[ \int \frac{6}{x} \, dx = 6 \int \frac{1}{x} \, dx = 6 \ln |x| \]

3. **Integrate \( 9 \)**:
\[ \int 9 \, dx = 9x \]

### Step 4: Combine the Results
Now we combine all the results from the integrations:
\[ \int \left( x^2 + \frac{6}{x} + 9 \right) dx = \frac{x^3}{3} + 6 \ln |x| + 9x + C \]
where \( C \) is the constant of integration.

### Final Answer
Thus, the final result of the integral is:
\[ \frac{x^3}{3} + 6 \ln |x| + 9x + C \]

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