Evaluate: `int_e^(e^2) (dx)/(x log x)`

Answer

To evaluate the integral \( I = \int_e^{e^2} \frac{dx}{x \log x} \), we will follow these steps:

### Step 1: Substitution
Let \( t = \log x \). Then, differentiating both sides gives us:
\[ \frac{dt}{dx} = \frac{1}{x} \implies dx = x \, dt \]
Since \( x = e^t \), we can substitute \( dx \) as:
\[ dx = e^t \, dt \]

### Step 2: Change the limits of integration
When \( x = e \):
\[ t = \log e = 1 \]
When \( x = e^2 \):
\[ t = \log e^2 = 2 \log e = 2 \]

### Step 3: Substitute in the integral
Now substituting \( x \) and \( dx \) into the integral:
\[ I = \int_{1}^{2} \frac{e^t \, dt}{e^t \cdot t} = \int_{1}^{2} \frac{dt}{t} \]

### Step 4: Evaluate the integral
The integral \( \int \frac{dt}{t} \) is:
\[ \int \frac{dt}{t} = \ln |t| \]
Thus, we evaluate it from 1 to 2:
\[ I = \left[ \ln t \right]_{1}^{2} = \ln 2 - \ln 1 \]

### Step 5: Simplify the result
Since \( \ln 1 = 0 \):
\[ I = \ln 2 - 0 = \ln 2 \]

### Final Answer
Thus, the value of the integral is:
\[ \boxed{\ln 2} \]

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