What is `inte^(e^(x))e^(x)` dx equal to ?

Options

A. `e^(e^(x))+C`
B. `2e^(e^(x))+C`
C. `e^(e^(x))e^(x)+C`
D. `2e^(e^(x))e^(x)+C`

Answer

To solve the integral \( \int e^{e^x} e^x \, dx \), we can follow these steps:

### Step-by-Step Solution:

1. **Identify the integral**:
We start with the integral:
\[ I = \int e^{e^x} e^x \, dx \]

2. **Substitution**:
We will use substitution to simplify the integral. Let:
\[ t = e^x \]
Then, the differential \( dt \) is given by:
\[ dt = e^x \, dx \quad \Rightarrow \quad dx = \frac{dt}{e^x} = \frac{dt}{t} \]

3. **Rewrite the integral**:
Substitute \( t \) and \( dx \) into the integral:
\[ I = \int e^t \cdot t \cdot \frac{dt}{t} = \int e^t \, dt \]

4. **Integrate**:
The integral of \( e^t \) is straightforward:
\[ I = e^t + C \]

5. **Back-substitute**:
Now, we substitute back \( t = e^x \):
\[ I = e^{e^x} + C \]

### Final Answer:
Thus, the result of the integral is:
\[ \int e^{e^x} e^x \, dx = e^{e^x} + C \]