What is `int e^(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`

Explanation

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

### Step 1: Substitution
Let \( t = e^x \). Then, we differentiate \( t \) with respect to \( x \):
\[ dt = e^x \, dx \quad \Rightarrow \quad dx = \frac{dt}{e^x} = \frac{dt}{t} \]

### Step 2: Rewrite the Integral
Now we can rewrite the integral in terms of \( t \):
\[ \int e^{e^x} e^x \, dx = \int e^t \cdot e^x \, dx = \int e^t \cdot t \cdot \frac{dt}{t} = \int e^t \, dt \]

### Step 3: Integrate
Now we can integrate \( e^t \):
\[ \int e^t \, dt = e^t + C \]

### Step 4: Substitute Back
Now we substitute back \( t = e^x \):
\[ e^t + C = e^{e^x} + C \]

### Final Answer
Thus, the integral \( \int e^{e^x} e^x \, dx \) is equal to:
\[ e^{e^x} + C \]

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