`int7^(7^(7^x))*7^(7^x)*7^x\ dx=`

Options

A. `(7^(7^(7^(x))))/((log_(e)7)^(3))+C`
B. `(7^(7^(7^(x))))/((log_(e)7)^(2))+C`
C. `7^(7^(7^(x))).(log7)^(3)+C`
D. none of these

Explanation

To solve the integral \( \int 7^{7^{7^x}} \cdot 7^{7^x} \cdot 7^x \, dx \), we will follow these steps:

### Step 1: Simplify the Integral
We can rewrite the integral as:
\[ \int 7^{7^{7^x}} \cdot 7^{7^x} \cdot 7^x \, dx = \int 7^{(7^{7^x} + 7^x + x)} \, dx \]

### Step 2: Substitution
Let \( t = 7^x \). Then, we differentiate to find \( dx \):
\[ \frac{dt}{dx} = 7^x \ln(7) \implies dx = \frac{dt}{t \ln(7)} \]

### Step 3: Rewrite the Integral in Terms of \( t \)
Now we substitute \( t \) into the integral:
\[ 7^{7^{7^x}} = 7^{7^t}, \quad 7^{7^x} = 7^t, \quad 7^x = t \]
Thus, the integral becomes:
\[ \int 7^{7^t + t} \cdot \frac{dt}{t \ln(7)} \]

### Step 4: Further Simplification
Now we can simplify the expression:
\[ = \frac{1}{\ln(7)} \int \frac{7^{7^t + t}}{t} \, dt \]

### Step 5: Second Substitution
Let \( u = 7^t \). Then, \( t = \log_7(u) \) and \( dt = \frac{du}{u \ln(7)} \). Substitute \( t \) and \( dt \):
\[ = \frac{1}{\ln(7)} \int \frac{7^{u}}{\log_7(u)} \cdot \frac{du}{u \ln(7)} \]

### Step 6: Integrate
Now we can integrate:
\[ = \frac{1}{(\ln(7))^2} \int 7^u \, du \]
The integral of \( 7^u \) is:
\[ = \frac{7^u}{\ln(7)} + C \]

### Step 7: Substitute Back
Now we substitute back \( u = 7^t \) and \( t = 7^x \):
\[ = \frac{7^{7^{7^x}}}{(\ln(7))^2 \ln(7)} + C \]
This simplifies to:
\[ = \frac{7^{7^{7^x}}}{(\ln(7))^3} + C \]

### Final Answer
Thus, the final result of the integral is:
\[ \int 7^{7^{7^x}} \cdot 7^{7^x} \cdot 7^x \, dx = \frac{7^{7^{7^x}}}{(\ln(7))^3} + C \]