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arc-cos-cos-x-1-2cos-x-dx-




Question Number 118081 by bemath last updated on 15/Oct/20
 ∫ arc cos (((cos x)/(1+2cos x))) dx =?
$$\:\int\:\mathrm{arc}\:\mathrm{cos}\:\left(\frac{\mathrm{cos}\:\mathrm{x}}{\mathrm{1}+\mathrm{2cos}\:\mathrm{x}}\right)\:\mathrm{dx}\:=? \\ $$
Commented by MJS_new last updated on 15/Oct/20
seems impossible to solve
$$\mathrm{seems}\:\mathrm{impossible}\:\mathrm{to}\:\mathrm{solve} \\ $$
Commented by MJS_new last updated on 16/Oct/20
this is one version of the Coxeter Integral  but it needs borders  ∫_0 ^(π/2) arccos ((cos x)/(1+2cos x)) dx=((5π^2 )/(24))  see the video or search the web
$$\mathrm{this}\:\mathrm{is}\:\mathrm{one}\:\mathrm{version}\:\mathrm{of}\:\mathrm{the}\:\mathrm{Coxeter}\:\mathrm{Integral} \\ $$$$\mathrm{but}\:\mathrm{it}\:\mathrm{needs}\:\mathrm{borders} \\ $$$$\underset{\mathrm{0}} {\overset{\frac{\pi}{\mathrm{2}}} {\int}}\mathrm{arccos}\:\frac{\mathrm{cos}\:{x}}{\mathrm{1}+\mathrm{2cos}\:{x}}\:{dx}=\frac{\mathrm{5}\pi^{\mathrm{2}} }{\mathrm{24}} \\ $$$$\mathrm{see}\:\mathrm{the}\:\mathrm{video}\:\mathrm{or}\:\mathrm{search}\:\mathrm{the}\:\mathrm{web} \\ $$
Commented by bemath last updated on 16/Oct/20
thank sir
$${thank}\:{sir} \\ $$
Answered by AbduraufKodiriy last updated on 15/Oct/20
Solution to this in YouTube channel  which is ′F lammable Maths′
$$\boldsymbol{{Solution}}\:\boldsymbol{{to}}\:\boldsymbol{{this}}\:\boldsymbol{{in}}\:\boldsymbol{{YouTube}}\:\boldsymbol{{channel}} \\ $$$$\boldsymbol{{which}}\:\boldsymbol{{is}}\:'\boldsymbol{{F}}\:\boldsymbol{{lammable}}\:\boldsymbol{{Maths}}' \\ $$

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