Evaluate : `int _(1) ^(e) ( dx)/( x (1 + log x))`

Answer

To evaluate the integral

\[ \int_{1}^{e} \frac{dx}{x(1 + \log x)}, \]

we will use substitution.

### Step 1: Substitution
Let \( t = \log x \). Then, we differentiate:

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

### Step 2: Change the limits of integration
When \( x = 1 \):
\[ t = \log(1) = 0. \]
When \( x = e \):
\[ t = \log(e) = 1. \]

### Step 3: Rewrite the integral
Now we can rewrite the integral in terms of \( t \):

\[ \int_{0}^{1} \frac{e^t \, dt}{e^t(1 + t)} = \int_{0}^{1} \frac{dt}{1 + t}. \]

### Step 4: Evaluate the integral
The integral

\[ \int \frac{dt}{1 + t} = \log(1 + t) + C. \]

Now we will evaluate it from \( 0 \) to \( 1 \):

\[ \left[ \log(1 + t) \right]_{0}^{1} = \log(1 + 1) - \log(1 + 0) = \log(2) - \log(1). \]

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

\[ \log(2) - 0 = \log(2). \]

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

\[ \log(2). \]

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