`int_(1)^(4) (2x^(2) +1) dx`

Answer

To solve the integral \( \int_{1}^{4} (2x^{2} + 1) \, dx \), we will break it down into steps.

### Step 1: Set up the integral
We start with the integral:
\[ I = \int_{1}^{4} (2x^{2} + 1) \, dx \]

### Step 2: Separate the integral
We can separate the integral into two parts:
\[ I = \int_{1}^{4} 2x^{2} \, dx + \int_{1}^{4} 1 \, dx \]

### Step 3: Integrate the first part
Using the power rule for integration, where \( \int x^{n} \, dx = \frac{x^{n+1}}{n+1} + C \):
\[ \int 2x^{2} \, dx = 2 \cdot \frac{x^{3}}{3} = \frac{2x^{3}}{3} \]
Now we evaluate this from 1 to 4:
\[ \left[ \frac{2x^{3}}{3} \right]_{1}^{4} = \frac{2(4^{3})}{3} - \frac{2(1^{3})}{3} \]
Calculating \( 4^{3} = 64 \):
\[ = \frac{2 \cdot 64}{3} - \frac{2 \cdot 1}{3} = \frac{128}{3} - \frac{2}{3} = \frac{126}{3} = 42 \]

### Step 4: Integrate the second part
Now we integrate the second part:
\[ \int 1 \, dx = x \]
Evaluating this from 1 to 4 gives:
\[ \left[ x \right]_{1}^{4} = 4 - 1 = 3 \]

### Step 5: Combine the results
Now we combine the results from both parts:
\[ I = 42 + 3 = 45 \]

### Final Answer
Thus, the value of the integral is:
\[ \int_{1}^{4} (2x^{2} + 1) \, dx = 45 \]

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