Mathematics - Definite Integration Question with Solution | TestHub

$\begin{array}{rl}& {\log }_4{\log }_6\left(3+4x-x^2\right)>0\\ & {\log }_6\left(3+4x-x^2\right)>1\end{array}$
$\begin{array}{rl}& 3+4x-x^2>6\\ & x^2-4x+3<0\\ & (x-1)(x-3)<0\\ & x\in (1,3)\\ & \text{ so }a=1\&b=3\\ & \Rightarrow {\int }_0^2\left[x^2\right]dx=?\\ & I={\int }_0^1\left[x^2\right]dx+{\int }_1^{\sqrt{2}}\left[x^2\right]dx+{\int }_{\sqrt{2}}^{\sqrt{3}}\left[x^2\right]dx+{\int }_{\sqrt{3}}^{\sqrt{4}}\left[x^2\right]dx\\ & =0+|x{|}_1^{\sqrt{2}}+2|x{|}_{\sqrt{2}}^{\sqrt{3}}+3|x{|}_{\sqrt{3}}^{\sqrt{4}}\end{array}$
$\begin{array}{rl}& =(\sqrt{2}-1)+2(\sqrt{3}-\sqrt{2})+3(2-\sqrt{3})\\ & =5-\sqrt{2}-\sqrt{3}\Rightarrow \mathrm{p}+\mathrm{q}+\mathrm{r}=10\end{array}$