Correct answer is 23
\(\int_{0}^{3}\left[x^{2}\right] d x+\int_{0}^{3}\left[\frac{x^{2}}{2}\right] d x\)
\(=\int_{0}^{1} 0 \mathrm{dx}+\int_{1}^{12} 1 \mathrm{dx}+\int_{\sqrt{2}}^{\sqrt{3}} 2 \mathrm{dx}\)
\(+\int_{\sqrt{3}}^{2} 3 \mathrm{~d} x+\int_{2}^{\sqrt{5}} 4 \mathrm{dx}+\int_{\sqrt{5}}^{\sqrt{6}} 5 \mathrm{dx}\)
\(+\int_{\sqrt{6}}^{\sqrt{7}} 6 \mathrm{dx}+\int_{\sqrt{7}}^{\sqrt{8}} 7 \mathrm{dx}+\int_{\sqrt{8}}^{3} 8 \mathrm{dx}\)
\(+\int_{0}^{\sqrt{2}} 0 \mathrm{dx}+\int_{\sqrt{2}}^{2} 1 \mathrm{dx}\)
\(+\int_{2}^{\sqrt{6}} 2 \mathrm{dx}+\int_{\sqrt{6}}^{\sqrt{8}} 3 \mathrm{dx}+\int_{\sqrt{8}}^{3} 4 \mathrm{dx}=31-6 \sqrt{2}-\sqrt{3}-\sqrt{5}\)
\(-2 \sqrt{6}-\sqrt{7}\)
\(a=31 \quad b=-6 \quad c=-2\)
\(a+b+c=31-6-2=23\)
