`inte^(e^(e^(x))).e^(e^(x)).e^(x)dx=....+C`

Options

A. `e^(e^(x))`
B. `(1)/(2)e^(x).e^(x)`
C. `e^(e^(e^(x)))`
D. `(1)/(2)e^(e^(x))`

Explanation

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

### Step 1: Simplify the Integral
Let:
\[ I = \int e^{e^{e^x}} \cdot e^{e^x} \cdot e^x \, dx \]

### Step 2: Substitution
We will use substitution to simplify the integral. Let:
\[ t = e^{e^x} \]
Then, we need to find \( dt \). First, we differentiate \( t \):
\[ \frac{dt}{dx} = e^{e^x} \cdot e^x \]
This implies:
\[ dt = e^{e^x} \cdot e^x \, dx \]

### Step 3: Rewrite the Integral
Now, we can rewrite the integral \( I \) in terms of \( t \):
\[ I = \int e^{t} \, dt \]

### Step 4: Integrate
The integral of \( e^t \) is:
\[ I = e^t + C \]

### Step 5: Substitute Back
Now, we substitute back \( t = e^{e^x} \):
\[ I = e^{e^{e^x}} + C \]

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

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