`int_(1//pi)^(2//pi)(sin(1//x))/(x^(2))dx=?`

Options

A. `1`
B. `(1)/(2)`
C. `(3)/(2)`
D. none of these

Explanation

To solve the integral \( \int_{\frac{1}{\pi}}^{\frac{2}{\pi}} \frac{\sin\left(\frac{1}{x}\right)}{x^2} \, dx \), we can use a substitution method. Here’s the step-by-step solution:

### Step 1: Substitution
Let \( t = \frac{1}{x} \). Then, we differentiate to find \( dx \):
\[ dx = -\frac{1}{t^2} dt \]
This means that \( \frac{1}{x^2} = t^2 \). Therefore, we can rewrite the integral as:
\[ \int \sin\left(\frac{1}{x}\right) \frac{1}{x^2} \, dx = \int \sin(t) \cdot t^2 \left(-\frac{1}{t^2}\right) dt = -\int \sin(t) \, dt \]

### Step 2: Change of Limits
Next, we need to change the limits of integration. When \( x = \frac{1}{\pi} \), then \( t = \pi \). When \( x = \frac{2}{\pi} \), then \( t = \frac{\pi}{2} \). Therefore, the limits change from \( x: \left[\frac{1}{\pi}, \frac{2}{\pi}\right] \) to \( t: \left[\pi, \frac{\pi}{2}\right] \).

### Step 3: Adjust the Integral
Now we can write the integral with the new limits:
\[ -\int_{\pi}^{\frac{\pi}{2}} \sin(t) \, dt \]
Since the limits are reversed, we can switch them and remove the negative sign:
\[ \int_{\frac{\pi}{2}}^{\pi} \sin(t) \, dt \]

### Step 4: Integrate
The integral of \( \sin(t) \) is:
\[ -\cos(t) \]
Thus, we evaluate:
\[ \left[-\cos(t)\right]_{\frac{\pi}{2}}^{\pi} \]

### Step 5: Evaluate the Limits
Now we substitute the limits:
\[ -\cos(\pi) - \left(-\cos\left(\frac{\pi}{2}\right)\right) = -(-1) - (0) = 1 \]

### Final Answer
Therefore, the value of the integral is:
\[ \int_{\frac{1}{\pi}}^{\frac{2}{\pi}} \frac{\sin\left(\frac{1}{x}\right)}{x^2} \, dx = 1 \]