Mathematics

$$\int _{ 0 }^{ 400\pi  }{ \sqrt { 1-\cos { 2x }  }  }$$


ANSWER

$$800\sqrt 2$$


SOLUTION
$$I=\int  _{ 0 }^{ 400\pi  }{ \sqrt { 1-\cos  x } dx } \\ =400\int  _{ 0 }^{ \pi  }{ \sqrt { 1-\cos  2x } dx } \\ =800\int  _{ 0 }^{ \pi  }{ \sqrt { 2 } \sqrt { { { \sin   }^{ 2 } }x } dx } \\ =800\sqrt { 2 } \int  _{ 0 }^{ \frac { \pi  }{ 2 }  }{ \sin  xdx } \\ =800\sqrt { 2 } \left[ { \cos  x } \right] _{ \frac { \pi  }{ 2 }  }^{ 0 } \\ =800\sqrt { 2 }$$

$$  \\ Hence,\, the\, option\, C\, is\, the\, correct\, answer. $$
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Single Correct Medium Published on 17th 09, 2020
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