The value of ∫_-π / 6^π / 6 ( π+4 x^11 1-sin (|x|+π / 6) ) d x is equal to:

Split: $\displaystyle \int_{-\pi/6}^{\pi/6} \frac{\pi}{1 - \sin(|x| + \pi/6)} dx + \int_{-\pi/6}^{\pi/6} \frac{4x^{11}}{1 - \sin(|x| + \pi/6)} dx$∫−π/6π/6​1−sin(∣x∣+π/6)π​dx+∫−π/6π/6​1−sin(∣x∣+π/6)4x11​dx.

Second integral = 0 (odd function over symmetric interval).

First integral: $\displaystyle 2\pi \int_{0}^{\pi/6} \frac{1}{1 - \sin(x + \pi/6)} dx$2π∫0π/6​1−sin(x+π/6)1​dx.

Let $u = x + \pi/6$u=x+π/6: $\displaystyle = 2\pi \int_{\pi/6}^{\pi/3} \frac{1}{1 - \sin u} du$=2π∫π/6π/3​1−sinu1​du.

$\frac{1}{1-\sin u} = \sec^2 u + \sec u \tan u$1−sinu1​=sec2u+secutanu.

$= 2\pi [\tan u + \sec u]_{\pi/6}^{\pi/3} = 2\pi[(\sqrt{3}+2) - \sqrt{3}] = 4\pi$=2π[tanu+secu]π/6π/3​=2π[(3​+2)−3​]=4π.