Key Concepts and Formulas

• Greatest Integer Function: The greatest integer function $[t]$ yields the largest integer less than or equal to $t$. For example, $[3.7] = 3$ and $[-2.3] = -3$.
• Definite Integration of Greatest Integer Function: To integrate a function of the form $\int_a^b [f(x)] dx$, we must identify the intervals within $[a, b]$ where $f(x)$ crosses integer values. These points, where $f(x) = k$ for some integer $k$, act as partition points for the integral. Within each sub-interval $[x_i, x_{i+1}]$, $[f(x)]$ remains a constant integer, say $m$. The integral over this sub-interval is then $\int_{x_i}^{x_{i+1}} m \, dx = m(x_{i+1} - x_i)$.
• Properties of Square Roots: Understanding the approximate values of common square roots like $\sqrt{2}$, $\sqrt{3}$, and $\sqrt{5}$ is useful for ordering them and determining the intervals.

Step-by-Step Solution

Step 1: Understand the Integral and the Function We are asked to evaluate the definite integral \int_\limits{0}^{2.4}\left[x^{2}\right] d x. The function inside the greatest integer bracket is $f(x) = x^2$. The limits of integration are from $0$ to $2.4$.

Step 2: Determine the Range of $x^2$ and Identify Critical Points As $x$ varies from $0$ to $2.4$, the value of $x^2$ varies from $0^2 = 0$ to $(2.4)^2 = 5.76$. Thus, $x^2 \in [0, 5.76]$. The value of $[x^2]$ changes when $x^2$ crosses an integer. The integers within the range $[0, 5.76]$ are $0, 1, 2, 3, 4, 5$. We need to find the values of $x$ in the interval $[0, 2.4]$ for which $x^2$ equals these integers. These $x$ values will serve as the points to split our integral:

• $x^2 = 0 \implies x = 0$
• $x^2 = 1 \implies x = 1$
• $x^2 = 2 \implies x = \sqrt{2}$
• $x^2 = 3 \implies x = \sqrt{3}$
• $x^2 = 4 \implies x = 2$
• $x^2 = 5 \implies x = \sqrt{5}$

We need to order these critical points and include the upper limit of integration, $2.4$. The approximate values are: $\sqrt{2} \approx 1.414$, $\sqrt{3} \approx 1.732$, $\sqrt{5} \approx 2.236$. All these values are within the interval $[0, 2.4]$.

Step 3: Split the Integral into Sub-intervals Based on the critical points identified in Step 2, we divide the interval $[0, 2.4]$ into sub-intervals where $[x^2]$ remains constant:

• For $x \in [0, 1)$, $x^2 \in [0, 1)$, so $[x^2] = 0$.
• For $x \in [1, \sqrt{2})$, $x^2 \in [1, 2)$, so $[x^2] = 1$.
• For $x \in [\sqrt{2}, \sqrt{3})$, $x^2 \in [2, 3)$, so $[x^2] = 2$.
• For $x \in [\sqrt{3}, 2)$, $x^2 \in [3, 4)$, so $[x^2] = 3$.
• For $x \in [2, \sqrt{5})$, $x^2 \in [4, 5)$, so $[x^2] = 4$.
• For $x \in [\sqrt{5}, 2.4]$, $x^2 \in [5, 5.76]$, so $[x^2] = 5$.

Now, we can rewrite the integral as a sum of integrals over these sub-intervals: $\int_{0}^{2.4}\left[x^{2}\right] d x = \int_{0}^{1}\left[x^{2}\right] d x+\int_{1}^{\sqrt{2}}\left[x^{2}\right] d x+\int_{\sqrt{2}}^{\sqrt{3}}\left[x^{2}\right] d x+\int_{\sqrt{3}}^{2}\left[x^{2}\right] d x+\int_{2}^{\sqrt{5}}\left[x^{2}\right] d x+\int_{\sqrt{5}}^{2.4}\left[x^{2}\right] d x$

Step 4: Evaluate Each Sub-integral Substitute the constant values of $[x^2]$ for each interval and perform the integration: $= \int_{0}^{1} 0 \, d x + \int_{1}^{\sqrt{2}} 1 \, d x + \int_{\sqrt{2}}^{\sqrt{3}} 2 \, d x + \int_{\sqrt{3}}^{2} 3 \, d x + \int_{2}^{\sqrt{5}} 4 \, d x + \int_{\sqrt{5}}^{2.4} 5 \, d x$ $= 0 \cdot (1-0) + 1 \cdot (\sqrt{2}-1) + 2 \cdot (\sqrt{3}-\sqrt{2}) + 3 \cdot (2-\sqrt{3}) + 4 \cdot (\sqrt{5}-2) + 5 \cdot (2.4-\sqrt{5})$

Step 5: Simplify the Resulting Expression Expand and collect like terms: $= 0 + (\sqrt{2}-1) + (2\sqrt{3}-2\sqrt{2}) + (6-3\sqrt{3}) + (4\sqrt{5}-8) + (12-5\sqrt{5})$ Group the constant terms, terms with $\sqrt{2}$, terms with $\sqrt{3}$, and terms with $\sqrt{5}$: Constant terms: $-1 + 6 - 8 + 12 = 9$ Terms with $\sqrt{2}$: $\sqrt{2} - 2\sqrt{2} = -\sqrt{2}$ Terms with $\sqrt{3}$: $2\sqrt{3} - 3\sqrt{3} = -\sqrt{3}$ Terms with $\sqrt{5}$: $4\sqrt{5} - 5\sqrt{5} = -\sqrt{5}$

Therefore, the value of the integral is: $9 - \sqrt{2} - \sqrt{3} - \sqrt{5}$

Step 6: Identify $\alpha, \beta, \gamma, \delta$ and Calculate Their Sum The problem states that the integral equals $\alpha+\beta \sqrt{2}+\gamma \sqrt{3}+\delta \sqrt{5}$. Comparing our result $9 - 1\sqrt{2} - 1\sqrt{3} - 1\sqrt{5}$ with this form, we get:

• $\alpha = 9$
• $\beta = -1$
• $\gamma = -1$
• $\delta = -1$

We need to find the sum $\alpha+\beta+\gamma+\delta$: $\alpha+\beta+\gamma+\delta = 9 + (-1) + (-1) + (-1) = 9 - 3 = 6$

Common Mistakes & Tips

• Forgetting the upper limit: Ensure that the upper limit of integration ($2.4$ in this case) is also considered when defining the last interval.
• Incorrectly determining intervals: Double-check that the values of $x^2$ are indeed constant integers within each chosen sub-interval.
• Algebraic errors: Be meticulous when expanding and combining terms, especially with signs and coefficients of the square roots.

Summary The problem involves integrating the greatest integer function of $x^2$. The strategy is to identify the points where $x^2$ crosses integer values within the given integration interval $[0, 2.4]$. These points are used to split the integral into several sub-integrals, each of which can be evaluated by treating $[x^2]$ as a constant. After performing the integrations and simplifying the resulting expression, we compare it to the given form $\alpha+\beta \sqrt{2}+\gamma \sqrt{3}+\delta \sqrt{5}$ to find the values of $\alpha, \beta, \gamma,$ and $\delta$. Finally, their sum is computed.

The final answer is $\boxed{6}$.