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 :
\[I=\int\limits_{0}^{\pi }{\sqrt{1+4{{\sin }^{2}}\frac{x}{2}-4\sin \frac{x}{2}dx}}\] |
\[=\int\limits_{0}^{\pi }{\left| 1-2\sin \frac{x}{2} \right|dx}\] |
\[=\int\limits_{0}^{\pi /3}{\left( 1-2\sin \frac{x}{2} \right)dx\int\limits_{\pi /3}^{\pi }{\left( 2\sin \frac{x}{2}-1 \right)dx}}\] |
\[=\left. \left( x+4\cos e\frac{x}{2} \right) \right|_{0}^{\pi /3}+\left. \left( -\,4\cos e\frac{\pi }{2}-x \right) \right|_{\pi /3}^{\pi }\] |
\[=\frac{\pi }{3}+8.\frac{\sqrt{3}}{2}-4\] |
\[=4\sqrt{3}-4-\frac{\pi }{3}.\] |
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