Evaluate `:-`
`int_(0)^(1)(3x^(2)+4)dx`

Explanation

To evaluate the integral \( I = \int_{0}^{1} (3x^2 + 4) \, dx \), we will follow these steps:

### Step 1: Break down the integral
We can separate the integral into two parts:
\[ I = \int_{0}^{1} 3x^2 \, dx + \int_{0}^{1} 4 \, dx \]

### Step 2: Evaluate the first integral
For the first integral, we need to find:
\[ \int 3x^2 \, dx \]
Using the power rule for integration, which states that:
\[ \int x^n \, dx = \frac{x^{n+1}}{n+1} + C \]
we can compute:
\[ \int 3x^2 \, dx = 3 \cdot \frac{x^{3}}{3} = x^3 \]
Now we will evaluate this from 0 to 1:
\[ \left[ x^3 \right]_{0}^{1} = 1^3 - 0^3 = 1 - 0 = 1 \]

### Step 3: Evaluate the second integral
Now we evaluate the second integral:
\[ \int 4 \, dx \]
The integral of a constant \( c \) is given by:
\[ \int c \, dx = cx + C \]
So,
\[ \int 4 \, dx = 4x \]
Now we will evaluate this from 0 to 1:
\[ \left[ 4x \right]_{0}^{1} = 4(1) - 4(0) = 4 - 0 = 4 \]

### Step 4: Combine the results
Now we combine the results of both integrals:
\[ I = 1 + 4 = 5 \]

### Final Answer
Thus, the value of the integral is:
\[ \boxed{5} \]