Question

Let $[t]$ denote the largest integer less than or equal to $t$. If \[ \int_0^1 \left(\left[x^2\right] + \left\lfloor \frac{x^2}{2} \right\rfloor\right) dx = a + b\sqrt{2} - \sqrt{3} - \sqrt{5} + c\sqrt{6} - \sqrt{7}, \] where $a, b, c \in \mathbb{Z}$, then $a + b + c$ is equal to ________.

Answer 1

Correct Answer: 23

The problem requests the evaluation of the definite integral \( \int_0^1 \left(\left[x^2\right] + \left\lfloor \frac{x^2}{2} \right\rfloor\right) dx \). For \(x \in [0, 1)\), \(x^2<1\) and \(x^2/2<1\), thus \([x^2] = 0\) and \(\lfloor x^2/2 \rfloor = 0\), leading to an integral value of 0. This contradicts the provided answer format. A common variant of this problem uses an upper limit of 3. Assuming the corrected problem is \( \int_0^3 \left(\left[x^2\right] + \left\lfloor \frac{x^2}{2} \right\rfloor\right) dx \), we proceed with this assumption to determine the value of \(a+b+c\).

Concept Used:

The greatest integer function, denoted by \([t]\) or \(\lfloor t \rfloor\), is a step function. The value of \(\lfloor f(x) \rfloor\) remains constant between points where \(f(x)\) evaluates to an integer. To integrate a step function, the interval of integration is divided into sub-intervals at these integer points. The integral is then calculated by summing the areas of rectangles formed over each sub-interval, where the height of each rectangle is the constant function value within that sub-interval.

Step-by-Step Solution:

Step 1: Split the integral into two separate integrals.

Let \(I = \int_0^3 \left(\left[x^2\right] + \left\lfloor \frac{x^2}{2} \right\rfloor\right) dx\). This can be expressed as:

\[I = \int_0^3 \left[x^2\right] dx + \int_0^3 \left\lfloor \frac{x^2}{2} \right\rfloor dx = I_1 + I_2\]

Step 2: Evaluate \(I_1 = \int_0^3 [x^2] dx\).

\([x^2]\) changes value when \(x^2\) is an integer. For \(x \in [0, 3]\), \(x^2\) ranges from 0 to 9. The points where \([x^2]\) changes are \(x = \sqrt{1}, \sqrt{2}, \sqrt{3}, \ldots, \sqrt{8}, 3\). We segment the integral at these points:

\[I_1 = \int_0^1 0\,dx + \int_1^{\sqrt{2}} 1\,dx + \int_{\sqrt{2}}^{\sqrt{3}} 2\,dx + \int_{\sqrt{3}}^{2} 3\,dx + \int_{2}^{\sqrt{5}} 4\,dx + \int_{\sqrt{5}}^{\sqrt{6}} 5\,dx + \int_{\sqrt{6}}^{\sqrt{7}} 6\,dx + \int_{\sqrt{7}}^{\sqrt{8}} 7\,dx + \int_{\sqrt{8}}^3 8\,dx\]

Step 3: Calculate the value of \(I_1\).

\[I_1 = 0 + (\sqrt{2}-1) + 2(\sqrt{3}-\sqrt{2}) + 3(2-\sqrt{3}) + 4(\sqrt{5}-2) + 5(\sqrt{6}-\sqrt{5}) + 6(\sqrt{7}-\sqrt{6}) + 7(2\sqrt{2}-\sqrt{7}) + 8(3-2\sqrt{2})\]\[= \sqrt{2}-1 + 2\sqrt{3}-2\sqrt{2} + 6-3\sqrt{3} + 4\sqrt{5}-8 + 5\sqrt{6}-5\sqrt{5} + 6\sqrt{7}-6\sqrt{6} + 14\sqrt{2}-7\sqrt{7} + 24-16\sqrt{2}\]

Combine like terms:

\[I_1 = (-1+6-8+24) + (1-2+14-16)\sqrt{2} + (2-3)\sqrt{3} + (4-5)\sqrt{5} + (5-6)\sqrt{6} + (6-7)\sqrt{7}\]\[I_1 = 21 - 3\sqrt{2} - \sqrt{3} - \sqrt{5} - \sqrt{6} - \sqrt{7}\]

Step 4: Evaluate \(I_2 = \int_0^3 \lfloor \frac{x^2}{2} \rfloor dx\).

\(\lfloor x^2/2 \rfloor\) changes value when \(x^2/2\) is an integer, meaning \(x^2 = 2k\). For \(x \in [0, 3]\), \(x^2/2\) ranges from 0 to 4.5. The points where \(\lfloor x^2/2 \rfloor\) changes are \(x = \sqrt{2}, \sqrt{4}, \sqrt{6}, \sqrt{8}\). We segment the integral accordingly:

\[I_2 = \int_0^{\sqrt{2}} 0\,dx + \int_{\sqrt{2}}^2 1\,dx + \int_2^{\sqrt{6}} 2\,dx + \int_{\sqrt{6}}^{\sqrt{8}} 3\,dx + \int_{\sqrt{8}}^3 4\,dx\]

Step 5: Calculate the value of \(I_2\).

\[I_2 = 0 + (2-\sqrt{2}) + 2(\sqrt{6}-2) + 3(2\sqrt{2}-\sqrt{6}) + 4(3-2\sqrt{2})\]\[= 2-\sqrt{2} + 2\sqrt{6}-4 + 6\sqrt{2}-3\sqrt{6} + 12-8\sqrt{2}\]

Combine like terms:

\[I_2 = (2-4+12) + (-1+6-8)\sqrt{2} + (2-3)\sqrt{6}\]\[I_2 = 10 - 3\sqrt{2} - \sqrt{6}\]

Step 6: Compute the total integral \(I = I_1 + I_2\).

\[I = (21 - 3\sqrt{2} - \sqrt{3} - \sqrt{5} - \sqrt{6} - \sqrt{7}) + (10 - 3\sqrt{2} - \sqrt{6})\]\[I = (21+10) + (-3-3)\sqrt{2} - \sqrt{3} - \sqrt{5} + (-1-1)\sqrt{6} - \sqrt{7}\]\[I = 31 - 6\sqrt{2} - \sqrt{3} - \sqrt{5} - 2\sqrt{6} - \sqrt{7}\]

Step 7: Match with the given expression to find \(a, b, c\).

The given result format is \(a + b\sqrt{2} - \sqrt{3} - \sqrt{5} + c\sqrt{6} - \sqrt{7}\).Comparing this with our calculated \(I\):

\[31 - 6\sqrt{2} - \sqrt{3} - \sqrt{5} - 2\sqrt{6} - \sqrt{7} = a + b\sqrt{2} - \sqrt{3} - \sqrt{5} + c\sqrt{6} - \sqrt{7}\]

By comparing coefficients, we obtain:

\[a = 31\]\[b = -6\]\[c = -2\]

The values \(a, b, c\) are all integers.

Finally, calculate \(a+b+c\):

\[a+b+c = 31 + (-6) + (-2) = 31 - 8 = 23\]

The value of \(a+b+c\) is 23.