Mathematics

Evaluate the given integral.
$$\displaystyle\int{\dfrac{\sqrt{1-\cos{2x}}}{2}}dx$$


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

$$=\displaystyle\int{\dfrac{\sqrt{1-1+2{\sin}^{2}{x}}}{2}}dx$$          ....$$(cos2x=1-2sin^2x)$$

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

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

$$=\dfrac{\sqrt{2}\times \sqrt{2}}{2\sqrt{2}}\times-\cos{x}+c$$

$$=\dfrac{-1}{\sqrt{2}}\cos{x}+c$$
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Subjective Medium Published on 17th 09, 2020
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