Evaluate the integrals : `I = int_(1//pi)^(2//pi) (sin""(1)/(x))/(x^(2)) dx`,

Answer

To evaluate the integral

\[ I = \int_{\frac{1}{\pi}}^{\frac{2}{\pi}} \frac{\sin\left(\frac{1}{x}\right)}{x^2} \, dx, \]

we will use substitution and properties of differentiation.

### Step 1: Substitution
Let \( t = \frac{1}{x} \). Then, we have:

\[ x = \frac{1}{t} \quad \text{and} \quad dx = -\frac{1}{t^2} \, dt. \]

### Step 2: Change the limits of integration
When \( x = \frac{1}{\pi} \),

\[ t = \frac{1}{\frac{1}{\pi}} = \pi. \]

When \( x = \frac{2}{\pi} \),

\[ t = \frac{1}{\frac{2}{\pi}} = \frac{\pi}{2}. \]

Thus, the limits change from \( x: \left[\frac{1}{\pi}, \frac{2}{\pi}\right] \) to \( t: \left[\pi, \frac{\pi}{2}\right] \).

### Step 3: Rewrite the integral
Substituting into the integral, we get:

\[ I = \int_{\pi}^{\frac{\pi}{2}} \sin(t) \left(-\frac{1}{t^2}\right) \left(-\frac{1}{t^2}\right) dt = \int_{\pi}^{\frac{\pi}{2}} \frac{\sin(t)}{t^2} \, dt. \]

### Step 4: Change the order of limits
Changing the limits of integration gives us:

\[ I = \int_{\frac{\pi}{2}}^{\pi} \frac{\sin(t)}{t^2} \, dt. \]

### Step 5: Evaluate the integral
Now we can evaluate the integral:

\[ I = -\left[-\cos(t)\right]_{\frac{\pi}{2}}^{\pi} = \left[-\cos(\pi) + \cos\left(\frac{\pi}{2}\right)\right]. \]

Calculating the values:

\[ \cos(\pi) = -1 \quad \text{and} \quad \cos\left(\frac{\pi}{2}\right) = 0. \]

Thus,

\[ I = -(-1) + 0 = 1. \]

### Final Answer
The value of the integral is

\[ \boxed{1}. \]

---