The value of `int5^(5^(5^(x))). 5^(5^(x)). 5^(x)` Is equal to

Options

A. `5^(5^(4))/(ln 5)^(3)+C`
B. `5^(5^(5^(5)))(ln 5)^(3)+C`
C. `5^(5^(5^(5)))/(ln 5)^(3)+C`
D. `5^(5^(5^(x)))/(ln 5)^(2)+C`

Answer

To solve the integral \( \int 5^{5^{5^x}} \cdot 5^{5^x} \cdot 5^x \, dx \), we can follow these steps:

### Step 1: Simplify the expression
We can combine the exponents in the integrand:
\[ 5^{5^{5^x}} \cdot 5^{5^x} \cdot 5^x = 5^{(5^{5^x} + 5^x + x)} \]

### Step 2: Substitution
Let’s make a substitution to simplify the integral. We can set:
\[ t = 5^{5^x} \]
Then, we need to find \( dt \) in terms of \( dx \).

### Step 3: Differentiate the substitution
To find \( dt \), we differentiate \( t \) with respect to \( x \):
\[ \frac{dt}{dx} = 5^{5^x} \cdot \ln(5) \cdot 5^x \]
This implies:
\[ dt = 5^{5^x} \cdot \ln(5) \cdot 5^x \, dx \]

### Step 4: Solve for \( dx \)
From the equation above, we can express \( dx \):
\[ dx = \frac{dt}{5^{5^x} \cdot \ln(5) \cdot 5^x} \]

### Step 5: Substitute in the integral
Now we substitute \( t \) and \( dx \) back into the integral:
\[ \int 5^{5^{5^x}} \cdot 5^{5^x} \cdot 5^x \, dx = \int t \cdot \frac{dt}{\ln(5) \cdot 5^{5^x} \cdot 5^x} \]
Since \( 5^{5^x} = t \), we can rewrite this as:
\[ \int t \cdot \frac{dt}{\ln(5) \cdot t \cdot 5^x} = \int \frac{1}{\ln(5) \cdot 5^x} \, dt \]

### Step 6: Integrate
Now we can integrate:
\[ = \frac{1}{\ln(5)} \int dt = \frac{t}{\ln(5)} + C \]

### Step 7: Substitute back for \( t \)
Finally, we substitute back \( t = 5^{5^x} \):
\[ = \frac{5^{5^{5^x}}}{\ln(5)} + C \]

### Final Answer
Thus, the value of the integral is:
\[ \int 5^{5^{5^x}} \cdot 5^{5^x} \cdot 5^x \, dx = \frac{5^{5^{5^x}}}{\ln(5)} + C \]