Question

${\int }_0^{\pi }|1+2\cos x|dx$ equals to

• A. $\frac{2\pi }{3}$
• B. $\pi$
• C. $2$
• D. $\frac{\pi }{3}+2\sqrt{3}$

Answer 1

The correct option is D $\frac{\pi }{3}+2\sqrt{3}$

$1+2\cos x\le 0$

$\cos x\le \frac{-1}{2}$

$x\ge \frac{2\pi }{3}$

$\Rightarrow {\int }_0^{\pi }|1+2\cos x|={\int }_0^{2\pi /3}1+2\cos x-{\int }_{2\pi /3}^{\pi }1+2\cos x$

$\Rightarrow {[x+2\sin x]}_0^{2\pi /3}-{[x+2\sin x]}_{2\pi /3}^{\pi }$

$\frac{2\pi }{3}+\sqrt{3}-(\pi -\frac{2\pi }{3}-\sqrt{3})$

$\frac{2\pi }{3}+\sqrt{3}-\pi +\frac{2\pi }{3}+\sqrt{3}$

$\Rightarrow \frac{\pi }{3}+2\sqrt{3}.$

Hence, the answer is $\frac{\pi }{3}+2\sqrt{3}.$