Integration (pi/3 to pi/2) root(1+cosx)/(1-cosx) 5/2 dx
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b=pi/3 a = pi/2
![\int\limits^a_b {\frac{\sqrt{1+cosx}}{(1-cosx)^\frac{5}{2}}} \, dx\\\\ = \int\limits^a_b {\frac{\sqrt2*Cos(\frac{x}{2})}{4\sqrt2*Sin^5\ \frac{x}{2}} \, dx \\\\=\frac{1}{4}[-\frac{2}{4}\frac{1}{Sin^4(x/2)}}]_b^a\\\\=\frac{-1}{8}[Cosec^4(\pi/2)-Cosec^4(\pi/3)]\\\\=\frac{7}{72}](https://tex.z-dn.net/?f=+%5Cint%5Climits%5Ea_b+%7B%5Cfrac%7B%5Csqrt%7B1%2Bcosx%7D%7D%7B%281-cosx%29%5E%5Cfrac%7B5%7D%7B2%7D%7D%7D+%5C%2C+dx%5C%5C%5C%5C+%3D+%5Cint%5Climits%5Ea_b+%7B%5Cfrac%7B%5Csqrt2%2ACos%28%5Cfrac%7Bx%7D%7B2%7D%29%7D%7B4%5Csqrt2%2ASin%5E5%5C+%5Cfrac%7Bx%7D%7B2%7D%7D+%5C%2C+dx+%5C%5C%5C%5C%3D%5Cfrac%7B1%7D%7B4%7D%5B-%5Cfrac%7B2%7D%7B4%7D%5Cfrac%7B1%7D%7BSin%5E4%28x%2F2%29%7D%7D%5D_b%5Ea%5C%5C%5C%5C%3D%5Cfrac%7B-1%7D%7B8%7D%5BCosec%5E4%28%5Cpi%2F2%29-Cosec%5E4%28%5Cpi%2F3%29%5D%5C%5C%5C%5C%3D%5Cfrac%7B7%7D%7B72%7D "\int\limits^a_b {\frac{\sqrt{1+cosx}}{(1-cosx)^\frac{5}{2}}} \, dx\\\\ = \int\limits^a_b {\frac{\sqrt2*Cos(\frac{x}{2})}{4\sqrt2*Sin^5\ \frac{x}{2}} \, dx \\\\=\frac{1}{4}[-\frac{2}{4}\frac{1}{Sin^4(x/2)}}]_b^a\\\\=\frac{-1}{8}[Cosec^4(\pi/2)-Cosec^4(\pi/3)]\\\\=\frac{7}{72}")
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