The value of the integral \[\int\limits_0^2 {\frac{{{{\left( {{\text{x}} – 1} \right)}^2}\sin \left( {{\text{x}} – 1} \right)}}{{{{\left( {{\text{x}} – 1} \right)}^2} + \cos \left( {{\text{x}} – 1} \right)}}} {\text{dx}}\] is A. 3 B. 0 C. -1 D. -2

The correct answer is $\boxed{\text{B) }0}$.

To solve this integral, we can use the following identity:

$$\int \frac{\sin(x)}{1+\cos(x)} \, dx = \text{Si}(x) + C$$

where $\text{Si}(x)$ is the sine integral function.

Substituting $x = x-1$, we get:

$$\int \frac{\sin(x-1)}{1+\cos(x-1)} \, dx = \text{Si}(x-1) + C$$

The limits of integration are $0$ and $2$. Substituting these in, we get:

$$\text{Si}(2) – \text{Si}(-1) = 0$$

Therefore, the value of the integral is $\boxed{\text{B) }0}$.

Here is a brief explanation of each option:

• Option A: $3$. This is the wrong answer because the integral does not equal $3$.
• Option B: $0$. This is the correct answer because the integral can be evaluated using the sine integral function.
• Option C: $-1$. This is the wrong answer because the integral does not equal $-1$.
• Option D: $-2$. This is the wrong answer because the integral does not equal $-2$.