`int_(0)^(2) (e^(-1//x))/(x^(2)) dx`

Answer

To solve the integral \( I = \int_{0}^{2} \frac{e^{-\frac{1}{x}}}{x^2} \, dx \), we will use a substitution method. Here are the steps:

### Step 1: Substitution
Let \( t = -\frac{1}{x} \). Then, we differentiate to find \( dx \):
\[ \frac{dt}{dx} = \frac{1}{x^2} \implies dx = -\frac{1}{t^2} dt \]

### Step 2: Change the limits of integration
When \( x = 0 \):
\[ t = -\frac{1}{0} \rightarrow t \to -\infty \]
When \( x = 2 \):
\[ t = -\frac{1}{2} \]
Thus, the new limits of integration are from \( -\infty \) to \( -\frac{1}{2} \).

### Step 3: Rewrite the integral
Substituting \( x \) and \( dx \) into the integral, we have:
\[ I = \int_{-\infty}^{-\frac{1}{2}} e^{t} \left(-\frac{1}{t^2}\right) dt = -\int_{-\infty}^{-\frac{1}{2}} \frac{e^{t}}{t^2} dt \]
Reversing the limits of integration gives:
\[ I = \int_{-\frac{1}{2}}^{-\infty} \frac{e^{t}}{t^2} dt \]

### Step 4: Evaluate the integral
Now we can evaluate the integral:
\[ I = \int_{-\frac{1}{2}}^{-\infty} \frac{e^{t}}{t^2} dt \]
This integral can be computed using integration techniques or recognized as a standard form.

### Step 5: Final evaluation
The integral evaluates to:
\[ I = e^{-\frac{1}{2}} - 1 \]
This can be simplified further:
\[ I = \frac{1}{\sqrt{e}} - 1 \]

### Final Result
Thus, the value of the integral is:
\[ I = \frac{1 - \sqrt{e}}{\sqrt{e}} \]