`int 5^(5^(5^x))*5^(5^x)*5^x *dx`

Options

A. `(5^(5x))/((log 5)^(3)) + c`
B. `5^(5^(5x))(log 5)^(3) + C`
C. `(5^(5^(5x)))/((log 5)^(3))+c`
D. None of these

Answer

To solve the integral \( I = \int 5^{5^{5^x}} \cdot 5^{5^x} \cdot 5^x \, dx \), we will use substitution and properties of exponents. Here’s a step-by-step solution:

### Step 1: Simplify the Integral
We can combine the exponents in the integrand:
\[ I = \int 5^{5^{5^x} + 5^{x} + x} \, dx \]

### Step 2: Use Substitution
Let’s make a substitution to simplify our integral. We can set:
\[ t = 5^{5^x} \]
Now, we need to find \( dt \):
\[ dt = 5^{5^x} \cdot \ln(5) \cdot 5^x \, dx \]
This implies:
\[ dx = \frac{dt}{5^{5^x} \cdot \ln(5) \cdot 5^x} \]

### Step 3: Rewrite the Integral in Terms of \( t \)
Substituting \( t \) into the integral, we have:
\[ I = \int t \cdot \frac{dt}{\ln(5) \cdot 5^x \cdot 5^{5^x}} \]
Notice that \( 5^x = 5^{\log_5(t)} = t^{\log_5(5)} = t \) and \( 5^{5^x} = t \). Therefore, we can rewrite the integral as:
\[ I = \int \frac{t^2}{\ln(5)} \cdot dt \]

### Step 4: Integrate
Now we can integrate:
\[ I = \frac{1}{\ln(5)} \cdot \frac{t^3}{3} + C \]
\[ I = \frac{t^3}{3 \ln(5)} + C \]

### Step 5: Substitute Back for \( t \)
Now, we substitute back for \( t \):
\[ t = 5^{5^x} \]
Thus, we have:
\[ I = \frac{(5^{5^x})^3}{3 \ln(5)} + C \]
\[ I = \frac{5^{3 \cdot 5^x}}{3 \ln(5)} + C \]

### Final Answer
The final result for the integral is:
\[ I = \frac{5^{3 \cdot 5^x}}{3 \ln(5)} + C \]