Integrate the following :
`int_(3)^(6)(u+at)dt` where u and a are constants.

Options

A. `3u+(27)a.`
B. `3u+(27)/(2)a.`
C. `3u+(25)/(2)a.`
D. `9u+(27)/(2)a.`

Explanation

To solve the definite integral \( \int_{3}^{6} (u + at) \, dt \), where \( u \) and \( a \) are constants, we can follow these steps:

### Step 1: Separate the Integral
We can separate the integral into two parts:
\[ \int_{3}^{6} (u + at) \, dt = \int_{3}^{6} u \, dt + \int_{3}^{6} at \, dt \]

### Step 2: Integrate the First Part
Since \( u \) is a constant, the integral of \( u \) with respect to \( t \) is:
\[ \int_{3}^{6} u \, dt = u \cdot \int_{3}^{6} dt = u \cdot [t]_{3}^{6} = u \cdot (6 - 3) = 3u \]

### Step 3: Integrate the Second Part
Now, we integrate \( at \):
\[ \int_{3}^{6} at \, dt = a \cdot \int_{3}^{6} t \, dt = a \cdot \left[ \frac{t^2}{2} \right]_{3}^{6} \]
Calculating the definite integral:
\[ = a \cdot \left( \frac{6^2}{2} - \frac{3^2}{2} \right) = a \cdot \left( \frac{36}{2} - \frac{9}{2} \right) = a \cdot \left( \frac{36 - 9}{2} \right) = a \cdot \frac{27}{2} \]

### Step 4: Combine the Results
Now, we combine the results from Step 2 and Step 3:
\[ \int_{3}^{6} (u + at) \, dt = 3u + a \cdot \frac{27}{2} \]

### Final Answer
Thus, the final result of the definite integral is:
\[ \int_{3}^{6} (u + at) \, dt = 3u + \frac{27a}{2} \]