Question

If \( [ ] \) denotes the greatest integer function, then \( \int_{1}^{2} [x^2] dx = \)

• A. \( 5 + \sqrt{2} + \sqrt{3} \)
• B. \( 5 + \sqrt{2} - \sqrt{3} \)
• C. \( 5 - \sqrt{2} - \sqrt{3} \)
• D. \( 5 - \sqrt{2} + \sqrt{3} \)

Hint:

For definite integrals involving \( [x^2] \), the critical points are always the square roots of integers. The result of such an integral can be visualized as the sum of areas of rectangles with heights equal to the integer values of the function.

Answer 1

The Correct Option is C

Concept: The Greatest Integer Function \( [f(x)] \) changes its value at points where \( f(x) \) becomes an integer. To integrate such a function, we must split the interval of integration at these critical points and evaluate the constant value of the function within each sub-interval.

Step 1: Identify the critical points for \( [x^2] \) in the interval \( [1, 2] \).
As \( x \) varies from \( 1 \) to \( 2 \), \( x^2 \) varies from \( 1^2 = 1 \) to \( 2^2 = 4 \). The integers between \( 1 \) and \( 4 \) are \( 1, 2, 3, \) and \( 4 \). The function \( [x^2] \) will change at:
• \( x^2 = 1 \Rightarrow x = 1 \)
• \( x^2 = 2 \Rightarrow x = \sqrt{2} \)
• \( x^2 = 3 \Rightarrow x = \sqrt{3} \)
• \( x^2 = 4 \Rightarrow x = 2 \)

Step 2: Split the integral based on these intervals.
We can rewrite the integral as: \[ I = \int_{1}^{\sqrt{2}} [x^2] dx + \int_{\sqrt{2}}^{\sqrt{3}} [x^2] dx + \int_{\sqrt{3}}^{2} [x^2] dx \]

Step 3: Evaluate the constant values of \( [x^2] \) in each sub-interval.

• For \( 1 \leq x < \sqrt{2} \), \( 1 \leq x^2 < 2 \), so \( [x^2] = 1 \).
• For \( \sqrt{2} \leq x < \sqrt{3} \), \( 2 \leq x^2 < 3 \), so \( [x^2] = 2 \).
• For \( \sqrt{3} \leq x < 2 \), \( 3 \leq x^2 < 4 \), so \( [x^2] = 3 \).

Step 4: Perform the integration.
\[ I = \int_{1}^{\sqrt{2}} 1 dx + \int_{\sqrt{2}}^{\sqrt{3}} 2 dx + \int_{\sqrt{3}}^{2} 3 dx \] \[ I = [x]_{1}^{\sqrt{2}} + [2x]_{\sqrt{2}}^{\sqrt{3}} + [3x]_{\sqrt{3}}^{2} \] \[ I = (\sqrt{2} - 1) + (2\sqrt{3} - 2\sqrt{2}) + (6 - 3\sqrt{3}) \] \[ I = \sqrt{2} - 1 + 2\sqrt{3} - 2\sqrt{2} + 6 - 3\sqrt{3} \] \[ I = 5 - \sqrt{2} - \sqrt{3} \]