Area of the region bounded by $y=x-[x],\ y=[x]$ and $\mathrm{x}$-axis in $[$0,2 $]$ is:

Trapezoidal rule for evaluation of $\displaystyle\int _{ a }^{ b }{ f(x)dx }$ requires the interval $(a,b)$ to be divided into

$\mathrm{P(x)=\frac{x^{2}}{x^{4}+1} \text { এবং } Q(x)=x^{2}}$

The value of $x$ such that $\displaystyle \int_{\sqrt{2}}^{\mathrm{x}}\frac{1}{x\sqrt{x^{2}-1}}dx=\frac{\pi}{12}$ is

$f(x)=x^{3}-3 x^{2}-2 x+1$

$y=\frac{1}{8} x^{2}$