Mathematics

Evaluate:

$$\int { \sqrt { \cfrac { 1-\cos { 2x }  }{ 2 }  }  } dx$$


SOLUTION
Given $$\displaystyle\int{\sqrt{\dfrac{1-\cos{2x}}{2}}dx}$$

$$=\dfrac{1}{\sqrt{2}}\displaystyle\int{\sqrt{1-\cos{2x}}dx}$$

$$=\dfrac{1}{\sqrt{2}}\displaystyle\int{\sqrt{1-\left(1-2{\sin}^{2}{x}\right)}dx}$$    $$[\because \cos 2x=1-2\sin^2 x]$$

$$=\dfrac{1}{\sqrt{2}}\displaystyle\int{\sqrt{2{\sin}^{2}{x}}dx}$$

$$=\dfrac{\sqrt{2}}{\sqrt{2}}\displaystyle\int{\sin{x}dx}$$

$$=\displaystyle\int{\sin{x}dx}$$

$$=-\cos{x}+c$$ 

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Subjective Medium Published on 17th 09, 2020
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