If `f(x)` in inegrable over `[1,2]` then `int_(1)^(2) f(x) dx` is equal to :

Options

A. `underset(nrarroo)"lim"(1)/(n)underset(r=1)overset(n)sumf(r/n)`
B. `underset(nrarroo)"lim"(1)/(n)underset(r=n+1)overset(2n)sumf(r/n)`
C. `underset(nrarroo)"lim"(1)/(n)underset(r=1)overset(n)sumf((r+n)/n)`
D. `underset(nrarroo)"lim"1/n underset(r=1)overset(2n)sumf(r/n)`

Answer

To solve the problem, we need to analyze the integrability of the function \( f(x) \) over the interval \([1, 2]\) and determine the value of the definite integral \( \int_{1}^{2} f(x) \, dx \).

### Step-by-Step Solution:

1. **Understanding Integrability**:
Since \( f(x) \) is integrable over the interval \([1, 2]\), it means that the definite integral \( \int_{1}^{2} f(x) \, dx \) exists and is a finite number.

**Hint**: Remember that a function is integrable on an interval if it is bounded and has a finite number of discontinuities.

2. **Setting Up the Integral**:
We need to evaluate \( \int_{1}^{2} f(x) \, dx \). This integral represents the area under the curve of \( f(x) \) from \( x=1 \) to \( x=2 \).

**Hint**: Visualizing the area under the curve can help understand the integral's meaning.

3. **Considering Options**:
The problem suggests checking different options for the integral. We will analyze the options to see which ones are equivalent to \( \int_{1}^{2} f(x) \, dx \).

4. **Evaluating Option B**:
For option B, we have:
\[ \lim_{n \to \infty} \frac{1}{n} \sum_{r=1}^{n} f\left(1 + \frac{r}{n}\right) \]
By substituting \( \frac{1}{n} = dx \) and \( \frac{r}{n} = x - 1 \), we can transform this sum into an integral:
\[ \int_{1}^{2} f(x) \, dx \]
This matches our original integral.

**Hint**: When converting sums to integrals, think about how Riemann sums approximate the area under a curve.

5. **Evaluating Option C**:
For option C, we have:
\[ \int_{1}^{2} f(x) \, dx \]
This is exactly the integral we are trying to evaluate. Therefore, this option is also correct.

**Hint**: Always check if the expression matches the integral you need to evaluate.

6. **Evaluating Other Options**:
- Option A and Option D do not represent the integral \( \int_{1}^{2} f(x) \, dx \) as they involve different limits or forms that do not correspond to the original integral.

**Hint**: Be cautious of the limits of integration; they determine the interval over which you are integrating.

### Conclusion:
The correct answers for the integral \( \int_{1}^{2} f(x) \, dx \) are options B and C.

**Final Answer**:
\[ \int_{1}^{2} f(x) \, dx \text{ is equal to the value obtained from options B and C.} \]