Question

The value of \( \int_0^\infty \left[ \sqrt{x} \right] dx \), where \( \left[ . \right] \) is the greatest integer function, is?

• A. \( 2 - \sqrt{2} \)
• B. \( \sqrt{2} - 2 \)
• C. \( 2 \)
• D. \( 2 \sqrt{2} \)

Hint:

When dealing with the greatest integer function, split the integral into intervals where the function is constant and integrate piecewise.

Answer 1

The Correct Option is C

The greatest integer function gives integer values, and the integral evaluates to 2 after solving by breaking it into intervals where the greatest integer remains constant.