JEE Main 2024 — Definite Integration Question with Solution

\int_\limits0^3\left(\left[x^2\right]+\left[\frac{x^2}{2}\right]\right) d x \quad=\int_\limits0^1 0 d x+\int_1^{\sqrt{2}} 1 d x+\int_\limits{\sqrt{2}}^{\sqrt{3}} 3 d x+

\begin{aligned} & \int_\limits{\sqrt{3}}^2 4 d x+\int_\limits2^{\sqrt{5}} 6 d x+\int_\limits{\sqrt{5}}^{\sqrt{6}} 7 d x+\int_\limits{\sqrt{6}}^{\sqrt{7}} 9 d x+\int_\limits{\sqrt{7}}^{\sqrt{8}} 10 d x+\int_\limits{\sqrt{8}}^3 12 d x \\ & =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{aligned}