Mathematics - Definite Integration Question with Solution | TestHub

Symmetry $x\mapsto \pi /2-x$ gives $2I=(\pi /2){\int }_0^{\pi /2}{\left[2+{\sin }^{2}2x-{\sin }^{4}2x\right]}^{-1}dx$. Partial fractions yield option (B). With $t=\tan x$, the two resulting rational integrals evaluate to give $I=\frac{\pi }{12}\left(1+\frac{2\pi }{3\sqrt{3}}\right)$, option (C).