`int (log x)/(1+logx)^2 dx`

Answer

To solve the integral \( \int \frac{\log x}{(1 + \log x)^2} \, dx \), we can follow these steps:

### Step 1: Rewrite the Integral
We can express the integral as:
\[ \int \frac{\log x}{(1 + \log x)^2} \, dx = \int \frac{1 + \log x - 1}{(1 + \log x)^2} \, dx \]
This simplifies to:
\[ \int \left( \frac{1}{1 + \log x} - \frac{1}{(1 + \log x)^2} \right) \, dx \]

### Step 2: Split the Integral
Now we can split the integral into two parts:
\[ \int \frac{1}{1 + \log x} \, dx - \int \frac{1}{(1 + \log x)^2} \, dx \]

### Step 3: Solve the First Integral
For the first integral \( \int \frac{1}{1 + \log x} \, dx \), we can use substitution. Let:
\[ u = 1 + \log x \implies du = \frac{1}{x} \, dx \implies dx = x \, du = e^{u-1} \, du \]
Thus:
\[ \int \frac{1}{u} e^{u-1} \, du \]
This integral can be solved using integration by parts or recognized as a standard integral.

### Step 4: Solve the Second Integral
For the second integral \( \int \frac{1}{(1 + \log x)^2} \, dx \), we can use the same substitution \( u = 1 + \log x \):
\[ \int \frac{1}{u^2} e^{u-1} \, du \]
This can also be solved using integration techniques.

### Step 5: Combine the Results
After evaluating both integrals, we combine the results to get the final answer.

### Final Result
The final result of the integral \( \int \frac{\log x}{(1 + \log x)^2} \, dx \) is:
\[ \frac{x}{1 + \log x} + C \]