If `int_(0)^(11) (11^(x))/(11^([x]))dx = k/(log11)`, (where [] denotes greatest integer function) then value of k is

Options

A. `11`
B. `101`
C. `110`
D. `121`

Answer

To solve the integral

\[ I = \int_{0}^{11} \frac{11^x}{11^{[x]}} \, dx \]

where \([x]\) denotes the greatest integer function, we can break the integral into segments based on the intervals defined by the greatest integer function.

### Step 1: Break the integral into intervals
The greatest integer function \([x]\) takes constant integer values within each interval. Thus, we can split the integral from \(0\) to \(11\) into segments:

\[ I = \int_{0}^{1} \frac{11^x}{11^0} \, dx + \int_{1}^{2} \frac{11^x}{11^1} \, dx + \int_{2}^{3} \frac{11^x}{11^2} \, dx + \ldots + \int_{10}^{11} \frac{11^x}{11^{10}} \, dx \]

This gives us:

\[ I = \int_{0}^{1} 11^x \, dx + \int_{1}^{2} \frac{11^x}{11} \, dx + \int_{2}^{3} \frac{11^x}{11^2} \, dx + \ldots + \int_{10}^{11} \frac{11^x}{11^{10}} \, dx \]

### Step 2: Evaluate each integral
We can evaluate each integral separately. The general form for the integral of \(11^x\) is:

\[ \int 11^x \, dx = \frac{11^x}{\log 11} \]

Now we can evaluate each segment:

1. For \( \int_{0}^{1} 11^x \, dx \):

\[ = \left[ \frac{11^x}{\log 11} \right]_{0}^{1} = \frac{11^1}{\log 11} - \frac{11^0}{\log 11} = \frac{11 - 1}{\log 11} = \frac{10}{\log 11} \]

2. For \( \int_{1}^{2} \frac{11^x}{11} \, dx \):

\[ = \frac{1}{11} \left[ \frac{11^x}{\log 11} \right]_{1}^{2} = \frac{1}{11} \left( \frac{11^2 - 11^1}{\log 11} \right) = \frac{1}{11} \cdot \frac{121 - 11}{\log 11} = \frac{110}{11 \log 11} = \frac{10}{\log 11} \]

3. Continuing this process, for \( \int_{2}^{3} \frac{11^x}{11^2} \, dx \):

\[ = \frac{1}{11^2} \left[ \frac{11^x}{\log 11} \right]_{2}^{3} = \frac{1}{121} \left( \frac{11^3 - 11^2}{\log 11} \right) = \frac{1}{121} \cdot \frac{1331 - 121}{\log 11} = \frac{1210}{121 \log 11} = \frac{10}{\log 11} \]

4. This pattern continues until \( \int_{10}^{11} \frac{11^x}{11^{10}} \, dx \):

\[ = \frac{1}{11^{10}} \left[ \frac{11^x}{\log 11} \right]_{10}^{11} = \frac{1}{11^{10}} \left( \frac{11^{11} - 11^{10}}{\log 11} \right) = \frac{1}{11^{10}} \cdot \frac{11^{10}(11 - 1)}{\log 11} = \frac{10}{\log 11} \]

### Step 3: Sum all integrals
Each integral contributes \(\frac{10}{\log 11}\) and there are \(11\) such segments (from \(0\) to \(11\)):

\[ I = 11 \cdot \frac{10}{\log 11} = \frac{110}{\log 11} \]

### Step 4: Relate to \(k\)
Given that

\[ I = \frac{k}{\log 11} \]

we can equate:

\[ \frac{110}{\log 11} = \frac{k}{\log 11} \]

Thus, we find:

\[ k = 110 \]

### Final Answer
The value of \(k\) is \(110\).