Evaluate the integrals : `I = int_(e)^(e^(2))(dx)/(x I n x)` ,

Answer

To evaluate the integral

\[ I = \int_{e}^{e^2} \frac{dx}{x \ln x}, \]

we will use substitution and properties of logarithms. Here are the steps:

### Step 1: Substitution
Let \( t = \ln x \). Then, differentiating both sides gives us:

\[ dt = \frac{1}{x} dx \quad \Rightarrow \quad dx = x \, dt = e^t \, dt. \]

### Step 2: Change the limits of integration
When \( x = e \),

\[ t = \ln e = 1. \]

When \( x = e^2 \),

\[ t = \ln e^2 = 2. \]

Thus, the limits of integration change from \( x = e \) to \( x = e^2 \) to \( t = 1 \) to \( t = 2 \).

### Step 3: Rewrite the integral
Substituting \( x \) and \( dx \) in terms of \( t \):

\[ I = \int_{1}^{2} \frac{e^t \, dt}{e^t \cdot t} = \int_{1}^{2} \frac{dt}{t}. \]

### Step 4: Evaluate the integral
Now, we can evaluate the integral:

\[ I = \int_{1}^{2} \frac{dt}{t} = \left[ \ln t \right]_{1}^{2} = \ln 2 - \ln 1. \]

Since \( \ln 1 = 0 \), we have:

\[ I = \ln 2 - 0 = \ln 2. \]

### Final Answer
Thus, the value of the integral is

\[ I = \ln 2. \]

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