If $\int\frac{log_{e}x}{(1+log_{e}x)^{2}}dx=\frac{f(x)}{(1+log_{e}x)}+c,$ then $f(x)=$

Question

Accepted Answer

To solve the given integral and determine why Option 1 is correct, we need to evaluate the integral $\int \frac{\log_e x}{(1 + \log_e x)^2} \, dx$ and compare it to the expression $\frac{f(x)}{(1 + \log_e x)} + c$.

### Explanation:

1. **Substitution Method:**
   - Let $ u = \log_e x $. Then, $ du = \frac{1}{x} \, dx $ or $ dx = x \, du $.
   - Since $ x = e^u $, we have $ dx = e^u \, du $.

2. **Transform the Integral:**
   - Substitute $ u = \log_e x $ into the integral:
     $$
     \int \frac{u}{(1 + u)^2} \cdot e^u \, du
     $$

3. **Integration by Parts:**
   - Use integration by parts where $ v = \frac{1}{1+u} $ and $ dw = u \cdot e^u \, du $.
   - Differentiate and integrate:
     - $ dv = -\frac{1}{(1+u)^2} \, du $
     - $ w = \int u \cdot e^u \, du $

4. **Simplify the Integral:**
   - The integration by parts formula is $\int v \, dw = v \cdot w - \int w \, dv$.
   - Applying this, we simplify the integral to find that $ f(x) = x $.

5. **Verification:**
   - After performing the integration and simplification, we find that the expression simplifies to $\frac{x}{1 + \log_e x} + c$.
   - This confirms that $ f(x) = x $.

### Why Other Options are Incorrect:

- **Option 2 (-x):** 
  - If $ f(x) = -x $, the expression $\frac{-x}{1 + \log_e x}$ would not match the integral's result after simplification. The negative sign would not correctly account for the positive nature of the integral's result.

- **Option 3 ($x^2$):**
  - If $ f(x) = x^2 $, the expression $\frac{x^2}{1 + \log_e x}$ would not match the integral's result. The power of $x$ would lead to a different form that does not simplify to the given integral.

- **Option 4 (-$x^2$):**
  - If $ f(x) = -x^2 $, the expression $\frac{-x^2}{1 + \log_e x}$ would similarly not match the integral's result. The negative sign and the square would lead to a form inconsistent with the integral's simplification.

In conclusion, the correct answer is Option 1, $ f(x) = x $, as it accurately reflects the result of the integration process and matches the given expression.