Let [t] denote the largest integer less than or equal to t.

Question

Let $[t]$ denote the largest integer less than or equal to $t$. If $\int _{0}^{3}([{x}^{2}]+[\frac{{x}^{2}}{2}])dx=a+b\sqrt{2}-\sqrt{3}-\sqrt{5}+c\sqrt{6}-\sqrt{7}$, where $a,b,c\in Z$, then $a+b+c$ is equal to \_\_\_\_\_\_\_\_\_\_.

Answer: 23

Accepted Answer

$\int _{0}^{3}([{x}^{2}]+[\frac{{x}^{2}}{2}])dx\, =\int _{0}^{1}0dx+{\int }_{\sqrt{2}}^{1}1dx+\int _{\sqrt{2}}^{\sqrt{3}}3dx+$

$\begin{gathered} \int _{\sqrt{3}}^{2}4dx+\int _{2}^{\sqrt{5}}6dx+\int _{\sqrt{5}}^{\sqrt{6}}7dx+\int _{\sqrt{6}}^{\sqrt{7}}9dx+\int _{\sqrt{7}}^{\sqrt{8}}10dx+\int _{\sqrt{8}}^{3}12dx \ =31-6\sqrt{2}-\sqrt{3}-\sqrt{5}-\sqrt{7}-2\sqrt{6} \ \Rightarrow a=31,b=-6,c=-2 \ \Rightarrow a+b+c=23 \end{gathered}$

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