The value of `int_(0)^(15)[x]^(3)dx` equals, where `[. ]` denote the greatest integer function

Options

A. 11005
B. 11025
C. 15625
D. None of these

Answer

To solve the integral \( \int_{0}^{15} [x]^3 \, dx \), where \([x]\) denotes the greatest integer function (GIF), we can break the integral into segments based on the intervals defined by the integers from 0 to 15.

### Step-by-Step Solution:

1. **Identify the intervals**: The greatest integer function \([x]\) changes its value at each integer. Therefore, we will break the integral from 0 to 15 into intervals:
\[ \int_{0}^{15} [x]^3 \, dx = \int_{0}^{1} [x]^3 \, dx + \int_{1}^{2} [x]^3 \, dx + \int_{2}^{3} [x]^3 \, dx + \ldots + \int_{14}^{15} [x]^3 \, dx \]

2. **Evaluate each interval**:
- For \(x \in [0, 1)\), \([x] = 0\):
\[ \int_{0}^{1} [x]^3 \, dx = \int_{0}^{1} 0^3 \, dx = 0 \]
- For \(x \in [1, 2)\), \([x] = 1\):
\[ \int_{1}^{2} [x]^3 \, dx = \int_{1}^{2} 1^3 \, dx = \int_{1}^{2} 1 \, dx = 1 \]
- For \(x \in [2, 3)\), \([x] = 2\):
\[ \int_{2}^{3} [x]^3 \, dx = \int_{2}^{3} 2^3 \, dx = \int_{2}^{3} 8 \, dx = 8 \]
- For \(x \in [3, 4)\), \([x] = 3\):
\[ \int_{3}^{4} [x]^3 \, dx = \int_{3}^{4} 3^3 \, dx = \int_{3}^{4} 27 \, dx = 27 \]
- For \(x \in [4, 5)\), \([x] = 4\):
\[ \int_{4}^{5} [x]^3 \, dx = \int_{4}^{5} 4^3 \, dx = \int_{4}^{5} 64 \, dx = 64 \]
- For \(x \in [5, 6)\), \([x] = 5\):
\[ \int_{5}^{6} [x]^3 \, dx = \int_{5}^{6} 5^3 \, dx = \int_{5}^{6} 125 \, dx = 125 \]
- For \(x \in [6, 7)\), \([x] = 6\):
\[ \int_{6}^{7} [x]^3 \, dx = \int_{6}^{7} 6^3 \, dx = \int_{6}^{7} 216 \, dx = 216 \]
- For \(x \in [7, 8)\), \([x] = 7\):
\[ \int_{7}^{8} [x]^3 \, dx = \int_{7}^{8} 7^3 \, dx = \int_{7}^{8} 343 \, dx = 343 \]
- For \(x \in [8, 9)\), \([x] = 8\):
\[ \int_{8}^{9} [x]^3 \, dx = \int_{8}^{9} 8^3 \, dx = \int_{8}^{9} 512 \, dx = 512 \]
- For \(x \in [9, 10)\), \([x] = 9\):
\[ \int_{9}^{10} [x]^3 \, dx = \int_{9}^{10} 9^3 \, dx = \int_{9}^{10} 729 \, dx = 729 \]
- For \(x \in [10, 11)\), \([x] = 10\):
\[ \int_{10}^{11} [x]^3 \, dx = \int_{10}^{11} 10^3 \, dx = \int_{10}^{11} 1000 \, dx = 1000 \]
- For \(x \in [11, 12)\), \([x] = 11\):
\[ \int_{11}^{12} [x]^3 \, dx = \int_{11}^{12} 11^3 \, dx = \int_{11}^{12} 1331 \, dx = 1331 \]
- For \(x \in [12, 13)\), \([x] = 12\):
\[ \int_{12}^{13} [x]^3 \, dx = \int_{12}^{13} 12^3 \, dx = \int_{12}^{13} 1728 \, dx = 1728 \]
- For \(x \in [13, 14)\), \([x] = 13\):
\[ \int_{13}^{14} [x]^3 \, dx = \int_{13}^{14} 13^3 \, dx = \int_{13}^{14} 2197 \, dx = 2197 \]
- For \(x \in [14, 15)\), \([x] = 14\):
\[ \int_{14}^{15} [x]^3 \, dx = \int_{14}^{15} 14^3 \, dx = \int_{14}^{15} 2744 \, dx = 2744 \]

3. **Combine all the results**:
\[ \int_{0}^{15} [x]^3 \, dx = 0 + 1 + 8 + 27 + 64 + 125 + 216 + 343 + 512 + 729 + 1000 + 1331 + 1728 + 2197 + 2744 \]

4. **Calculate the total**:
\[ = 0 + 1 + 8 + 27 + 64 + 125 + 216 + 343 + 512 + 729 + 1000 + 1331 + 1728 + 2197 + 2744 = 11025 \]

Thus, the value of \( \int_{0}^{15} [x]^3 \, dx \) is \( 11025 \).