Let the domain of the function $f(\mathrm{x})=\log _2 \log _4 \log _6\left(3+4 x-x^2\right)$ be $(\mathrm{a}, \mathrm{~b})$. If $\int_0^{\mathrm{b}-\mathrm{a}}\left[\mathrm{x}^2\right] \mathrm{dx}=\mathrm{p}-\sqrt{\mathrm{q}}-\sqrt{\mathrm{r}}, \mathrm{p}, \mathrm{q},$ $\mathrm{r} \in \mathbb{N},…

Question

Let the domain of the function $f(\mathrm{x})=\log _2 \log _4 \log _6\left(3+4 x-x^2\right)$ be $(\mathrm{a}, \mathrm{~b})$. If $\int_0^{\mathrm{b}-\mathrm{a}}\left[\mathrm{x}^2\right] \mathrm{dx}=\mathrm{p}-\sqrt{\mathrm{q}}-\sqrt{\mathrm{r}}, \mathrm{p}, \mathrm{q},$ $\mathrm{r} \in \mathbb{N}, \operatorname{gcd}(\mathrm{p}, \mathrm{q}, \mathrm{r})=1,$ where [$\cdot]$ is the greatest integer function, then $\mathrm{p}+\mathrm{q}+\mathrm{r}$ is equal to

Aicharya · Accepted Answer

$\begin{aligned} & \log _4 \log _6\left(3+4 x-x^2\right) \gt 0 \\ & \log _6\left(3+4 x-x^2\right) \gt 1\end{aligned}$
$\begin{aligned} & 3+4 x-x^2 \gt 6 \\ & x^2-4 x+3 \lt 0 \\ & (x-1)(x-3) \lt 0 \\ & x \in(1,3) \\ & \text { so } a=1 \& b=3 \\ & \Rightarrow \int_0^2\left[x^2\right] d x=? \\ & I=\int_0^1\left[x^2\right] d x+\int_1^{\sqrt{2}}\left[x^2\right] d x+\int_{\sqrt{2}}^{\sqrt{3}}\left[x^2\right] d x+\int_{\sqrt{3}}^{\sqrt{4}}\left[x^2\right] d x \\ & =0+|x|_1^{\sqrt{2}}+2|x|_{\sqrt{2}}^{\sqrt{3}}+3|x|_{\sqrt{3}}^{\sqrt{4}}\end{aligned}$
$\begin{aligned} & =(\sqrt{2}-1)+2(\sqrt{3}-\sqrt{2})+3(2-\sqrt{3}) \\ & =5-\sqrt{2}-\sqrt{3} \Rightarrow \mathrm{p}+\mathrm{q}+\mathrm{r}=10\end{aligned}$

Let the domain of the function $f(\mathrm{x})=\log _2 \log _4 \log _6\left(3+4 x-x^2\right)$ be $(\mathrm{a}…

Solution