Integrate the following function
`int_(o)^(2) 2t dt`

Options

A. `4`
B. `2`
C. `3`
D. `1`

Answer

To solve the integral \( \int_{0}^{2} 2t \, dt \), we will follow these steps:

### Step 1: Set up the integral
We start with the integral:
\[ \int_{0}^{2} 2t \, dt \]

### Step 2: Integrate the function
The integral of \( 2t \) with respect to \( t \) can be calculated as follows:
\[ \int 2t \, dt = 2 \cdot \frac{t^2}{2} = t^2 \]

### Step 3: Apply the limits of integration
Now we will evaluate the integral from the lower limit (0) to the upper limit (2):
\[ \left[ t^2 \right]_{0}^{2} = (2^2) - (0^2) \]

### Step 4: Calculate the values
Calculating the values gives:
\[ = 4 - 0 = 4 \]

### Final Answer
Thus, the value of the integral \( \int_{0}^{2} 2t \, dt \) is:
\[ \boxed{4} \]

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