A) \[\pi -4\]
B) \[\frac{2\pi }{3}-4-4\sqrt{3}\]
C) \[4\sqrt{3}-4\]
D) \[4\sqrt{3}-4-\frac{\pi }{3}\]
Correct Answer: D
Solution :
\[I=\int\limits_{0}^{\pi }{\sqrt{1+4{{\sin }^{2}}\frac{x}{2}-4\sin \frac{x}{2}}}\] \[I=\int\limits_{0}^{\pi }{\left| 1-2\sin \frac{x}{2} \right|dx}\] \[I=\int\limits_{0}^{\frac{\pi }{3}}{\left( 1-2\sin \frac{x}{2} \right)}dx+I=\int\limits_{\frac{\pi }{3}}^{\frac{\pi }{\pi }}{\left( 1-2\sin \frac{x}{2} \right)dx}\] \[I=\left[ x+4\cos \frac{x}{2} \right]_{0}^{\frac{\pi }{3}}+\left[ -x-4\cos \frac{x}{2} \right]_{-\frac{\pi }{3}}^{\pi }\] \[I=\frac{\pi }{3}+4\left( \frac{\sqrt{3}}{2}-1 \right)-\left( \pi -\frac{\pi }{3} \right)-4\left( 0-\frac{\sqrt{3}}{2} \right)\] \[I=\frac{\pi }{3}+2\sqrt{3}-4-\frac{2\pi }{3}+2\sqrt{3}=4\sqrt{3}-4-\frac{\pi }{3}\]
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