In the expansion off `(1+x)^(10)=.^(10)C_(0)+.^(10)C_(1)x+.^(10)C_(2)x^(2)+ . . .+.^(10)C_(10)x^(10)`, then value of
`528[(.^(10)C_(0))/(2)-(.^(10)C_(1))/(3)+(.^(10)C_(2))/(4)-(.^(10)C_(3))/(5)+ . . .+(.^(10)C_(10))/(12)]` is equal to________.

Answer

To solve the given problem, we need to evaluate the expression:

\[ 528 \left[ \frac{{\binom{10}{0}}}{2} - \frac{{\binom{10}{1}}}{3} + \frac{{\binom{10}{2}}}{4} - \frac{{\binom{10}{3}}}{5} + \ldots + \frac{{\binom{10}{10}}}{12} \right] \]

### Step 1: Understanding the Series
The series can be represented as:

\[ S = \sum_{k=0}^{10} (-1)^k \frac{\binom{10}{k}}{k+2} \]

### Step 2: Using Integration
To evaluate this sum, we can use the integral representation of the series. The integral representation for the sum can be derived from the binomial expansion:

\[ \int_0^1 x(1+x)^{10} \, dx \]

### Step 3: Calculate the Integral
We can simplify the integral:

\[ \int_0^1 x(1+x)^{10} \, dx = \int_0^1 x \sum_{k=0}^{10} \binom{10}{k} x^k \, dx = \sum_{k=0}^{10} \binom{10}{k} \int_0^1 x^{k+1} \, dx \]

The integral \(\int_0^1 x^{k+1} \, dx\) evaluates to \(\frac{1}{k+2}\). Thus, we have:

\[ \int_0^1 x(1+x)^{10} \, dx = \sum_{k=0}^{10} \binom{10}{k} \frac{1}{k+2} \]

### Step 4: Evaluating the Integral
Now we need to evaluate the integral:

\[ \int_0^1 x(1+x)^{10} \, dx \]

Using integration by parts, let \(u = (1+x)^{10}\) and \(dv = x \, dx\). Then \(du = 10(1+x)^9 \, dx\) and \(v = \frac{x^2}{2}\).

Using integration by parts:

\[ \int u \, dv = uv - \int v \, du \]

Calculating this gives:

\[ \int_0^1 x(1+x)^{10} \, dx = \left[ \frac{x^2}{2}(1+x)^{10} \right]_0^1 - \int_0^1 \frac{x^2}{2} \cdot 10(1+x)^9 \, dx \]

Evaluating the boundary terms gives:

\[ = \frac{1}{2}(1+1)^{10} - 0 = \frac{1}{2} \cdot 1024 = 512 \]

Now we need to evaluate the second integral, which can be simplified further.

### Step 5: Final Calculation
After evaluating the integral, we find:

\[ \int_0^1 x(1+x)^{10} \, dx = \frac{512}{12} \]

Thus, we can substitute back into our original expression:

\[ S = \frac{512}{12} \]

Now substituting this into our original expression:

\[ 528 \cdot S = 528 \cdot \frac{512}{12} = 44 \cdot 512 = 22528 \]

### Final Answer
The value of the expression is:

\[ \boxed{22528} \]