`int_(0)^(3)(x^(2)+1)dx`

Answer

To solve the integral \( \int_{0}^{3} (x^{2} + 1) \, dx \), we will follow these steps:

### Step 1: Set up the integral
We start with the integral:
\[ I = \int_{0}^{3} (x^{2} + 1) \, dx \]

### Step 2: Split the integral
We can split the integral into two separate integrals:
\[ I = \int_{0}^{3} x^{2} \, dx + \int_{0}^{3} 1 \, dx \]

### Step 3: Integrate \( x^{2} \)
Using the power rule for integration, where \( \int x^{n} \, dx = \frac{x^{n+1}}{n+1} + C \), we find:
\[ \int x^{2} \, dx = \frac{x^{3}}{3} \]
Now, we evaluate this from 0 to 3:
\[ \int_{0}^{3} x^{2} \, dx = \left[ \frac{x^{3}}{3} \right]_{0}^{3} = \frac{3^{3}}{3} - \frac{0^{3}}{3} = \frac{27}{3} - 0 = 9 \]

### Step 4: Integrate \( 1 \)
The integral of 1 is simply:
\[ \int 1 \, dx = x \]
Now, we evaluate this from 0 to 3:
\[ \int_{0}^{3} 1 \, dx = \left[ x \right]_{0}^{3} = 3 - 0 = 3 \]

### Step 5: Combine the results
Now we combine the results from the two integrals:
\[ I = 9 + 3 = 12 \]

### Final Answer
Thus, the value of the integral is:
\[ \boxed{12} \]

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