`int3^(3^(3^(x)))3^(3^(x))3^(x)dx` is equal to

Options

A. `(x^(3^(x)))/((log3)^(3))+C`
B. `3^(3^(3^(x)))(log3)^(3)+C`
C. `(3^(3^(3^(x))))/((log3)^(3))+C`
D. none of these

Answer

To solve the integral \( I = \int 3^{3^{3^x}} \cdot 3^{3^x} \cdot 3^x \, dx \), we can follow these steps:

### Step 1: Rewrite the Integral
We start with the integral:
\[ I = \int 3^{3^{3^x}} \cdot 3^{3^x} \cdot 3^x \, dx \]
We can combine the exponents since they have the same base:
\[ I = \int 3^{3^{3^x} + 3^{x} + x} \, dx \]

### Step 2: Substitution
Let \( t = 3^{3^x} \). Then, we need to find \( dt \):
Taking the logarithm of both sides:
\[ \log t = 3^x \log 3 \]
Differentiating both sides gives:
\[ \frac{dt}{dx} = 3^x \log 3 \cdot \log 3 = 3^x (\log 3)^2 \]
Thus, we can express \( dx \) in terms of \( dt \):
\[ dx = \frac{dt}{3^x (\log 3)^2} \]

### Step 3: Substitute Back into the Integral
Now we substitute \( t \) and \( dx \) into the integral:
\[ I = \int 3^{t} \cdot \frac{dt}{3^x (\log 3)^2} \]
We know that \( 3^x = \frac{t}{\log 3} \), so we can rewrite the integral:
\[ I = \int \frac{3^{t}}{t / \log 3} \cdot \frac{dt}{(\log 3)^2} \]
This simplifies to:
\[ I = \frac{1}{(\log 3)^3} \int 3^t \, dt \]

### Step 4: Integrate
The integral of \( 3^t \) is:
\[ \int 3^t \, dt = \frac{3^t}{\log 3} + C \]
Substituting back, we have:
\[ I = \frac{1}{(\log 3)^3} \left( \frac{3^t}{\log 3} + C \right) \]
Substituting \( t = 3^{3^x} \) back in:
\[ I = \frac{3^{3^{3^x}}}{(\log 3)^4} + C \]

### Final Answer
Thus, the final result of the integral is:
\[ I = \frac{3^{3^{3^x}}}{(\log 3)^4} + C \]