Question #41188

Let the vectors $\vec{a}(x)$, $\vec{b}(x)$, and $\vec{c}(x)$ be defined as: $\vec{a}(x) = \sin^{-1}(\sin x)\hat{i} + \cos^{-1}(\cos x)\hat{j} + \tan^{-1}(\tan x)\hat{k}$ $\vec{b}(x) = \cos^{-1}(\cos x)\hat{i} + \tan^{-1}(\tan x)\hat{j} + \sin^{-1}(\sin x)\hat{k}$ $\vec{c}(x) = \tan^{-1}(\tan x)\hat{i} + \sin^{-1}(\sin x)\hat{j} + \cos^{-1}(\cos x)\hat{k}$ For $x \in \left(\frac{\pi}{2}, \pi\right)$, let $\vec{a}', \vec{b}', \vec{c}'$ be the reciprocal system of vectors and $V'(x)$ be the volume of the parallelepiped formed by them. Evaluate the integral: $I = \int_{\pi/2}^{\pi} \frac{1}{x V'(x)} dx$