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December 30th, 2017, 03:05 AM   #1
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integrate root(cos2x)/sinx

integrate root(cos2x)/sinx
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December 30th, 2017, 08:19 AM   #2
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integrate root(cos2x)/sinx
Wolfram|Alpha: Computational Knowledge Engine)
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December 30th, 2017, 07:04 PM   #3
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Two approaches:
  1. Weierstrauss substitution;
  2. $\displaystyle \frac{\sqrt{\cos{(2x)}}}{\sin x} = \sin x \frac{\sqrt{2\cos^2 x - 1}}{1-\cos^2 x}$
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December 30th, 2017, 09:47 PM   #4
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Assuming $|x| \leqslant \pi/4$,

$\displaystyle \frac12\ln\left(\frac{2\cos(x) + 1 - \sqrt{\cos(2x)}}{2\cos(x) - 1 + \sqrt{\cos(2x)}}\right) + \sqrt2\ln\left(\sqrt{2\cos(2x)} + 2\cos(x)\right) + \ln\!|\!\tan(x/2)| + \text{C}$, where $\text{C}$ is a constant.
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