Question

If \([x]\) is the greatest integer function, then evaluate the integral \( \int_{0}^{5} [x] \, dx \):

• A. 15
• B. 2
• C. 3
• D. 10

Hint:

When dealing with the greatest integer function in integrals, consider the changes at integer boundaries to simplify your calculations.

Answer 1

The Correct Option is D

To solve the integral \( \int_{0}^{5} [x] \, dx \), we break it into intervals where \([x]\) remains constant: \[ \int_{0}^{1} [x] \, dx = \int_{0}^{1} 0 \, dx = 0, \\ \int_{1}^{2} [x] \, dx = \int_{1}^{2} 1 \, dx = 1, \\ \int_{2}^{3} [x] \, dx = \int_{2}^{3} 2 \, dx = 2, \\ \int_{3}^{4} [x] \, dx = \int_{3}^{4} 3 \, dx = 3, \\ \int_{4}^{5} [x] \, dx = \int_{4}^{5} 4 \, dx = 4. \] Adding these up: \[ 0 + 1 + 2 + 3 + 4 = 10. \]