The range of `f(x)=[1+sinx]+[2+si n x/2]+[3+si n x/3]++[n+si n x/n]AAx in [0,pi]` , where [.] denotes the greatest integer function, is, `{(n+n-2^2)/2,(n(n+1))/2}` `{(n(n+1))/2}` `{(n(n+1))/2, (n^2+n+2)/2}` `[(n(n+1))/2,(n^2+n+2)/2]`

Options

A. `{(n ^(2) +n-2)/(2), (n(n+1))/(2)}`
B. `{(n (n+1))/(2)}`
C. `{(n(n +1))/(2), (n ^(2) +n+2)/(2), (n ^(2)+ n+4)/(2)}`
D. `{(n (n+1))/(2), (n ^(2) +n-2)/(2)}`

Answer

To find the range of the function \( f(x) = [1 + \sin x] + [2 + \sin \frac{x}{2}] + [3 + \sin \frac{x}{3}] + \ldots + [n + \sin \frac{x}{n}] \) for \( x \in [0, \pi] \), where \([.]\) denotes the greatest integer function, we can follow these steps:

### Step 1: Understand the Function Components
The function consists of a sum of greatest integer functions. Each term in the function can be expressed as:
\[ f(x) = [k + \sin \frac{x}{k}] \quad \text{for } k = 1, 2, \ldots, n \]
where \( \sin \frac{x}{k} \) varies as \( x \) changes from \( 0 \) to \( \pi \).

### Step 2: Analyze the Range of Each Term
The sine function varies between \( 0 \) and \( 1 \) for \( x \in [0, \pi] \). Therefore, each term \( [k + \sin \frac{x}{k}] \) can take on the following values:
- When \( \sin \frac{x}{k} = 0 \), \( [k + 0] = k \).
- When \( \sin \frac{x}{k} = 1 \), \( [k + 1] = k + 1 \).

Thus, each term \( [k + \sin \frac{x}{k}] \) can take on the values \( k \) or \( k + 1 \).

### Step 3: Calculate the Minimum Value of \( f(x) \)
The minimum value occurs when all sine terms are zero:
\[ f_{\text{min}}(x) = [1 + 0] + [2 + 0] + [3 + 0] + \ldots + [n + 0] = 1 + 2 + 3 + \ldots + n = \frac{n(n + 1)}{2} \]

### Step 4: Calculate the Maximum Value of \( f(x) \)
The maximum value occurs when all sine terms are at their maximum:
\[ f_{\text{max}}(x) = [1 + 1] + [2 + 1] + [3 + 1] + \ldots + [n + 1] = 2 + 3 + 4 + \ldots + (n + 1) \]
This can be expressed as:
\[ f_{\text{max}}(x) = \left(1 + 2 + 3 + \ldots + n\right) + n = \frac{n(n + 1)}{2} + n = \frac{n(n + 1) + 2n}{2} = \frac{n^2 + 3n}{2} \]

### Step 5: Determine the Range of \( f(x) \)
The possible values of \( f(x) \) therefore range from:
\[ \frac{n(n + 1)}{2} \quad \text{to} \quad \frac{n^2 + 3n}{2} \]

### Conclusion
The range of \( f(x) \) is:
\[ \left[\frac{n(n + 1)}{2}, \frac{n^2 + 3n}{2}\right] \]

### Final Answer
The correct option is:
\[ \left[\frac{n(n + 1)}{2}, \frac{n^2 + n + 2}{2}\right] \]