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 $-2$.