The value of _- /6^ /6( +4x^111- (|x|+ /6))dx is equal to:

Question

The value of $\int _{-\pi /6}^{\pi /6}(\frac{\pi +4{x}^{11}}{1-\sin (|x|+\pi /6)})dx$ is equal to:

Accepted Answer

$\begin{gathered} {\int }_{\frac{\pi }{6}}^{0}\frac{\pi +4{x}^{11}}{1-\sin (|x|+\frac{\pi }{6})}+\frac{\pi -4{x}^{11}}{1-\sin (|x|+\frac{\pi }{6})}dx \ =2\pi {\int }_{\frac{\pi }{6}}^{0}\frac{1}{1-\sin (x+\frac{\pi }{6})}dx \ =2\pi {\int }_{\frac{\pi }{6}}^{0}\frac{1+\sin (x+\frac{\pi }{6})}{{\cos }^{2}(x+\frac{\pi }{6})}dx \ =2\pi {\int }_{\frac{\pi }{6}}^{0}{\sec }^{2}(x+\frac{\pi }{6})+	an (x+\frac{\pi }{6})\sec (x+\frac{\pi }{6})dx \ \Rightarrow 2\pi {[	an (x+\frac{\pi }{6})+\sec (x+\frac{\pi }{6})]}_{\frac{\pi }{6}}^{0} \ \Rightarrow 4\pi \end{gathered}$

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