integrate square root of 1-sin2xdx , [o, pie/4]
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integrate square root of 1-sin2xdx , [o, pie/4]
Hello, twilightstr!
Under the radical, multiply by: $\displaystyle \frac{1+\sin2x}{1+\sin2x}$Quote:
$\displaystyle \int^{\frac{\pi}{4}}_0 \sqrt{1-\sin2x}\,dx$
. . $\displaystyle \sqrt{\frac{1-\sin2x}{1}\cdot\frac{1+\sin2x}{1+\sin2x}} \;=\;\sqrt{\frac{1-\sin^2\!2x}{1+\sin2x}} \;=\;\sqrt{\frac{\cos^2\!2x}{1 + \sin2x}}\;=\;\frac{\cos2x}{\sqrt{1+\sin2x}} $
And we have: .$\displaystyle \int^{\frac{\pi}{4}}_0 (1 + \sin2x)^{-\frac{1}{2}}(\cos2x\,dx)$
Then let: $\displaystyle u \:=\:1 + \sin2x$
Got it?
$\displaystyle \sqrt{{{\cos }^{2}}2x}=\left| \cos 2x \right|=\cos 2x,$ because this last is positive for $\displaystyle \left[ 0,\frac{\pi }{4} \right].$
Another way is to recognize that:
$\displaystyle 1 - \sin{(2x)} = \sin^2{x} - 2\sin{x}\cos{x} + \cos^2{x} = (\sin{x} - \cos{x})^2$
Also take note that:
$\displaystyle \sqrt{(\sin{x} - \cos{x})^2} = |\sin{x} - \cos{x}| = -(\sin{x}-\cos{x})$ for $\displaystyle \left[ 0, \frac{\pi}{4} \right]$
