The value of `int_(e)^(pi^(2))[log_(pi)x] d(log_(e)x) ` (where [.] denotes greatest integer function) is

Options

A. `2log_(e)pi`
B. `log_(e)pi`
C. 1
D. 0

Explanation

To solve the integral \( I = \int_{e}^{\pi^2} [\log_{\pi} x] \, d(\log_e x) \), where \([.]\) denotes the greatest integer function, we will follow these steps:

### Step 1: Change of Variable
Let \( t = \log_e x \). Then, we have:
\[ x = e^t \quad \text{and} \quad d(\log_e x) = dt \]

### Step 2: Change the Limits of Integration
When \( x = e \):
\[ t = \log_e e = 1 \]
When \( x = \pi^2 \):
\[ t = \log_e \pi^2 = 2 \log_e \pi \]
Thus, the integral becomes:
\[ I = \int_{1}^{2 \log_e \pi} [\log_{\pi}(e^t)] \, dt \]

### Step 3: Simplify the Logarithm
Using the change of base formula for logarithms, we have:
\[ \log_{\pi}(e^t) = t \cdot \log_{\pi} e \]
Thus, the integral can be rewritten as:
\[ I = \int_{1}^{2 \log_e \pi} [t \cdot \log_{\pi} e] \, dt \]

### Step 4: Determine the Greatest Integer Function
Let \( c = \log_{\pi} e \). The integral now is:
\[ I = \int_{1}^{2 \log_e \pi} [ct] \, dt \]

### Step 5: Determine the Range of \( ct \)
- At \( t = 1 \):
\[ c \cdot 1 = \log_{\pi} e \]
- At \( t = 2 \log_e \pi \):
\[ c \cdot (2 \log_e \pi) = 2 \log_e \pi \cdot \log_{\pi} e = 2 \]

### Step 6: Analyze the Greatest Integer Function
The value of \( c \) is positive. Since \( \log_{\pi} e \) is less than 1 (because \( e < \pi \)), we have:
\[ \log_{\pi} e < 1 \quad \text{and} \quad 2 \log_{\pi} e < 2 \]
Thus, the greatest integer function \( [ct] \) will take the value 0 when \( t \) is between 1 and \( \log_{\pi} e \) and will take the value 1 when \( t \) is between \( \log_{\pi} e \) and \( 2 \log_{\pi} e \).

### Step 7: Split the Integral
We can split the integral based on the ranges:
\[ I = \int_{1}^{\log_{\pi} e} 0 \, dt + \int_{\log_{\pi} e}^{2 \log_{\pi} e} 1 \, dt \]

### Step 8: Evaluate the Integral
The first integral evaluates to 0. The second integral evaluates to:
\[ \int_{\log_{\pi} e}^{2 \log_{\pi} e} 1 \, dt = [t]_{\log_{\pi} e}^{2 \log_{\pi} e} = 2 \log_{\pi} e - \log_{\pi} e = \log_{\pi} e \]

### Final Result
Thus, the value of the integral is:
\[ I = \log_{\pi} e \]