If `int2^(2^(x)).2^(x)dx=A.2^(2^(x))+c`, then A =

Options

A. `(1)/(log2)`
B. `log2`
C. `(log2)^(2)`
D. `(1)/((log2)^(2))`

Answer

To solve the integral \( \int 2^{2^x} \cdot 2^x \, dx \) and find the constant \( A \) such that

\[ \int 2^{2^x} \cdot 2^x \, dx = A \cdot 2^{2^x} + c, \]

we will follow these steps:

### Step 1: Simplify the Integral
First, we can rewrite the integral:

\[ \int 2^{2^x} \cdot 2^x \, dx = \int 2^{2^x + x} \, dx. \]

### Step 2: Use Substitution
Let \( t = 2^x \). Then, we differentiate \( t \) with respect to \( x \):

\[ \frac{dt}{dx} = 2^x \ln(2) \implies dx = \frac{dt}{2^x \ln(2)} = \frac{dt}{t \ln(2)}. \]

### Step 3: Substitute in the Integral
Now, substitute \( t \) and \( dx \) into the integral:

\[ \int 2^{2^x + x} \, dx = \int 2^{t + \log_2(t)} \cdot \frac{dt}{t \ln(2)}. \]

Since \( 2^{\log_2(t)} = t \), we can simplify:

\[ 2^{t + \log_2(t)} = 2^t \cdot t. \]

Thus, the integral becomes:

\[ \int \frac{2^t \cdot t}{t \ln(2)} \, dt = \frac{1}{\ln(2)} \int 2^t \, dt. \]

### Step 4: Integrate \( 2^t \)
The integral of \( 2^t \) is:

\[ \int 2^t \, dt = \frac{2^t}{\ln(2)} + C. \]

### Step 5: Substitute Back
Now, substituting back for \( t \):

\[ \int 2^{2^x} \cdot 2^x \, dx = \frac{1}{\ln(2)} \cdot \left( \frac{2^t}{\ln(2)} + C \right) = \frac{2^{2^x}}{(\ln(2))^2} + C. \]

### Step 6: Compare with Given Expression
We have:

\[ \int 2^{2^x} \cdot 2^x \, dx = \frac{2^{2^x}}{(\ln(2))^2} + C. \]

From the problem statement, we know:

\[ A \cdot 2^{2^x} + c. \]

By comparing the coefficients, we find:

\[ A = \frac{1}{(\ln(2))^2}. \]

### Step 7: Final Answer
Thus, the value of \( A \) is:

\[ A = \frac{1}{(\ln(2))^2}. \]