The value of integral _-pi / 6^pi / 6(fracpi+4 x^111-sin (|x|+pi / 6)) d x is equal to

=2 pi integral _0^pi / 6 frac11-sin (x+(pi / 6)) d x let x+(x / 6)=t d x=d t =2 pi integral _pi / 6^pi / 3 frac dt1-sin t =2 pi integral _pi / 6^pi / 3 frac1+sin tcos ^2 t dt =2 pi[integral _pi / 6^pi / 3 sec ^2 tdt+integral _pi / 6^pi / 3 sec t tan tdt] =2 pi[(tan t)_pi / 6^pi / 3+(sec t)_pi / 6^pi / 3] =2 pi[(sqrt(3)-frac1sqrt(3))+(2-frac2sqrt(3))] =2 pi[sqrt(3)+2-sqrt(3)]=4 pi

The value of int pi 6 pi 6 frac pi 4 x 111 sin x pi 6 d... | null (Mathematics)

=2 \pi \int_{0}^{\pi / 6} \frac{1}{1-\sin \left(x+\frac{\pi}{6}\right)} d x\\

let

x+\frac{x}{6}=t d x=d t\\

=2 \pi \int_{\pi / 6}^{\pi / 3} \frac{\mathrm{dt}}{1-\sin \mathrm{t}} \\=2 \pi \int_{\pi / 6}^{\pi / 3} \frac{1+\sin \mathrm{t}}{\cos ^{2} \mathrm{t}} \mathrm{dt}\\

=2 \pi\left[\int_{\pi / 6}^{\pi / 3} \sec ^{2} \mathrm{tdt}+\int_{\pi / 6}^{\pi / 3} \sec \mathrm{t} \tan \mathrm{tdt}\right]\\

=2 \pi\left[(\tan \mathrm{t})_{\pi / 6}^{\pi / 3}+(\sec \mathrm{t})_{\pi / 6}^{\pi / 3}\right]\\

=2 \pi\left[\left(\sqrt{3}-\frac{1}{\sqrt{3}}\right)+\left(2-\frac{2}{\sqrt{3}}\right)\right]\\

=2 \pi[\sqrt{3}+2-\sqrt{3}]=4 \pi

Explanation

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