Question

If $I={\int }_{-\pi /6}^{\pi /6}\frac{\pi +4x^5}{1-\sin (\left|x\right|+\pi /6)}dx,$ then $I$ equals

• A. $4\pi$
• B. $2\pi +1\sqrt{3}$
• C. $2\pi -\sqrt{3}$
• D. $4\pi +\sqrt{3}-1/\sqrt{3}$

Answer 1

The correct option is A $4\pi$

As $\frac{4x^5}{1-\sin (\left|x\right|+\pi /6)}$ is an odd function, and $\frac{\pi }{1-\sin (\left|x\right|+\pi /6)}$ is an even function, we get
$I=2\pi {\int }_0^{\pi /6}\frac{dx}{1-\sin (x+\pi /6)},$
Substitute $x+\pi /6=t,dx=dt$
$I=2\pi {\int }_{-\pi /6}^{\pi /3}\frac{dt}{1-\sin t}=2\pi {\int }_{\pi /6}^{\pi /3}\frac{1+\sin t}{{\cos }^{2}t}dt$
$=2\pi (\tan t+\sec t){]}_{\pi /6}^{\pi /3}$
$=2\pi \left\{\right(\sqrt{3}+2)-(\frac{1}{\sqrt{3}}+\frac{2}{\sqrt{3}})\}=4\pi$
Ans: A