`int a^(a^(a^x)).a^(a^x) .a^(x) dx` is equal to

Options

A. `(a^(a^x))/(("log" a)^3) + c`
B. `a^(a^(a^x)) ("log" a)^3 + c`
C. `(a^(a^(a^x)))/(("log" a)^3) + c`
D. none of these

Answer

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

### Step 1: Simplify the Integral
We start with the integral:
\[ I = \int a^{(a^{(a^x)})} \cdot a^{(a^x)} \cdot a^x \, dx \]
This can be rewritten as:
\[ I = \int a^{(a^{(a^x)} + a^x + x)} \, dx \]

### Step 2: Use Substitution
Let us make the substitution:
\[ t = a^{(a^x)} \]
Then, we differentiate \( t \) with respect to \( x \):
\[ \frac{dt}{dx} = a^{(a^x)} \cdot \log(a) \cdot a^x \cdot \log(a) \]
Thus, we have:
\[ dt = a^{(a^x)} \cdot a^x \cdot (\log a)^2 \, dx \]
From this, we can express \( dx \) in terms of \( dt \):
\[ dx = \frac{dt}{a^{(a^x)} \cdot a^x \cdot (\log a)^2} \]

### Step 3: Substitute in the Integral
Now substituting \( t \) and \( dx \) into the integral:
\[ I = \int a^t \cdot \frac{dt}{a^{(a^x)} \cdot a^x \cdot (\log a)^2} \]
This simplifies to:
\[ I = \frac{1}{(\log a)^2} \int a^t \, dt \]

### Step 4: Integrate \( a^t \)
The integral of \( a^t \) is:
\[ \int a^t \, dt = \frac{a^t}{\log a} + C \]
Thus, we have:
\[ I = \frac{1}{(\log a)^2} \left( \frac{a^t}{\log a} + C \right) \]

### Step 5: Substitute Back for \( t \)
Now substituting back \( t = a^{(a^x)} \):
\[ I = \frac{1}{(\log a)^3} a^{(a^{(a^x)})} + C \]

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