question_answer

Find the value of \[\int\limits_{-\pi /3}^{\pi /3}{\frac{\pi +4{{x}^{3}}}{2-\cos \,\,\left( |x|+\frac{\pi }{3} \right)}dx}\]

A)  \[\frac{4\pi }{\sqrt{3}}\,\,\left[ {{\tan }^{-1}}3-\frac{\pi }{4} \right]\]      

B)  \[\frac{4\pi }{\sqrt{3}}\,\,\left[ {{\tan }^{-1}}2-\frac{\pi }{4} \right]\]

C)  \[\frac{4\pi }{\sqrt{5}}\,\,\left[ {{\tan }^{-1}}3-\frac{\pi }{2} \right]\]                  

D)  \[\frac{4\pi }{\sqrt{5}}\,\,\left[ {{\tan }^{-1}}3-\frac{\pi }{6} \right]\]

E)  None of these