Find the value of `int_(u)^(v)Mvdv`

Answer

To solve the integral \( \int_{u}^{v} M v \, dv \), we can follow these steps:

### Step 1: Identify the integral
We need to evaluate the integral:
\[ \int_{u}^{v} M v \, dv \]
where \( M \) is a constant (mass in this context).

### Step 2: Factor out the constant
Since \( M \) is a constant, we can factor it out of the integral:
\[ M \int_{u}^{v} v \, dv \]

### Step 3: Integrate \( v \)
Now we need to integrate \( v \):
\[ \int v \, dv = \frac{v^2}{2} \]
Thus, we can write:
\[ M \int_{u}^{v} v \, dv = M \left[ \frac{v^2}{2} \right]_{u}^{v} \]

### Step 4: Evaluate the definite integral
Now we evaluate the definite integral from \( u \) to \( v \):
\[ M \left[ \frac{v^2}{2} - \frac{u^2}{2} \right] \]

### Step 5: Simplify the result
This simplifies to:
\[ M \left( \frac{v^2}{2} - \frac{u^2}{2} \right) = \frac{M}{2} (v^2 - u^2) \]

### Final Result
Thus, the value of the integral \( \int_{u}^{v} M v \, dv \) is:
\[ \frac{M}{2} (v^2 - u^2) \]