The indefinite integral `inte^(e^(x))((xe^(x).lnx+1)/(x))dx` simplifies to (where, c is the constant of integration)

Options

A. `x ln (lnx)+c`
B. `e^(e^(x))+c`
C. `e^(e^(x))/(x)+c`
D. `e^(e^(x)). Ln x+c`

Answer

To solve the indefinite integral

\[ I = \int e^{e^x} \left( \frac{x e^x \ln x + 1}{x} \right) dx, \]

we can simplify and solve it step by step.

### Step 1: Simplify the Integral
We can rewrite the integral as:

\[ I = \int e^{e^x} \left( e^x \ln x + \frac{1}{x} \right) dx. \]

### Step 2: Distribute the Integral
This can be separated into two integrals:

\[ I = \int e^{e^x} e^x \ln x \, dx + \int e^{e^x} \frac{1}{x} \, dx. \]

### Step 3: Recognize the Structure
Let’s denote:

- \( u = e^x \) so that \( du = e^x \, dx \).
- The first integral becomes:

\[ \int e^{u} \ln(\ln u) \, du. \]

### Step 4: Apply Integration by Parts
For the first integral, we can use integration by parts. Let:

- \( v = \ln(\ln u) \) and \( dv = \frac{1}{\ln u} \cdot \frac{1}{u} \, du \).
- \( dw = e^u \, du \) and \( w = e^u \).

Using integration by parts:

\[ \int v \, dw = vw - \int w \, dv. \]

### Step 5: Substitute and Simplify
After applying integration by parts, we get:

\[ I = e^{e^x} \ln(\ln(e^x)) - \int e^{e^x} \cdot \frac{1}{\ln(e^x)} \cdot \frac{1}{e^x} \, dx + C. \]

### Step 6: Combine Results
The second integral simplifies to:

\[ \int e^{e^x} \frac{1}{x} \, dx. \]

Combining both parts gives us:

\[ I = e^{e^x} \ln x + C. \]

### Final Result
Thus, the indefinite integral simplifies to:

\[ I = e^{e^x} \ln x + C. \]

### Conclusion
The correct answer is:

\[ \text{Option D: } e^{e^x} \ln x + C. \]