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Question Number 151638 by mathdanisur last updated on 22/Aug/21

∫_( 0) ^( 2𝛑) (1 - cosx)^(10)  cos(10x) dx = ?

$$\underset{\:\mathrm{0}} {\overset{\:\mathrm{2}\boldsymbol{\pi}} {\int}}\left(\mathrm{1}\:-\:\mathrm{cos}\boldsymbol{\mathrm{x}}\right)^{\mathrm{10}} \:\mathrm{cos}\left(\mathrm{10x}\right)\:\mathrm{dx}\:=\:? \\ $$

Answered by Olaf_Thorendsen last updated on 22/Aug/21

I = ∫_0 ^(2Ο€) (1βˆ’cosx)^(10) cos(10x) dx  I = Re∫_0 ^(2Ο€) (1βˆ’cosx)^(10) e^(10ix)  dx  I = Re∫_0 ^(2Ο€) (1βˆ’((e^(ix) +e^(βˆ’ix) )/2))^(10) e^(10ix)  dx  Let u = e^(ix)   I = Re∫_(∣u∣=1) (1βˆ’((u+(1/u))/2))^(10) u^(10)  (βˆ’i(du/u))  I = (1/2^(10) )Re∫_(∣u∣=1) βˆ’(i/u)(uβˆ’1)^(20)  du  I = (1/2^(10) )Re(2iΟ€(Res(βˆ’(i/u)(uβˆ’1)^(20) ,0))  I = (1/2^(10) )Re(2iπ×(βˆ’i))  I = ((2Ο€)/2^(10) ) = (Ο€/2^9 ) = (Ο€/(512))

$$\mathrm{I}\:=\:\int_{\mathrm{0}} ^{\mathrm{2}\pi} \left(\mathrm{1}βˆ’\mathrm{cos}{x}\right)^{\mathrm{10}} \mathrm{cos}\left(\mathrm{10}{x}\right)\:{dx} \\ $$$$\mathrm{I}\:=\:\mathrm{Re}\int_{\mathrm{0}} ^{\mathrm{2}\pi} \left(\mathrm{1}βˆ’\mathrm{cos}{x}\right)^{\mathrm{10}} {e}^{\mathrm{10}{ix}} \:{dx} \\ $$$$\mathrm{I}\:=\:\mathrm{Re}\int_{\mathrm{0}} ^{\mathrm{2}\pi} \left(\mathrm{1}βˆ’\frac{{e}^{{ix}} +{e}^{βˆ’{ix}} }{\mathrm{2}}\right)^{\mathrm{10}} {e}^{\mathrm{10}{ix}} \:{dx} \\ $$$$\mathrm{Let}\:{u}\:=\:{e}^{{ix}} \\ $$$$\mathrm{I}\:=\:\mathrm{Re}\int_{\mid{u}\mid=\mathrm{1}} \left(\mathrm{1}βˆ’\frac{{u}+\frac{\mathrm{1}}{{u}}}{\mathrm{2}}\right)^{\mathrm{10}} {u}^{\mathrm{10}} \:\left(βˆ’{i}\frac{{du}}{{u}}\right) \\ $$$$\mathrm{I}\:=\:\frac{\mathrm{1}}{\mathrm{2}^{\mathrm{10}} }\mathrm{Re}\int_{\mid{u}\mid=\mathrm{1}} βˆ’\frac{{i}}{{u}}\left({u}βˆ’\mathrm{1}\right)^{\mathrm{20}} \:{du} \\ $$$$\mathrm{I}\:=\:\frac{\mathrm{1}}{\mathrm{2}^{\mathrm{10}} }\mathrm{Re}\left(\mathrm{2}{i}\pi\left(\mathrm{Res}\left(βˆ’\frac{{i}}{{u}}\left({u}βˆ’\mathrm{1}\right)^{\mathrm{20}} ,\mathrm{0}\right)\right)\right. \\ $$$$\mathrm{I}\:=\:\frac{\mathrm{1}}{\mathrm{2}^{\mathrm{10}} }\mathrm{Re}\left(\mathrm{2}{i}\piΓ—\left(βˆ’{i}\right)\right) \\ $$$$\mathrm{I}\:=\:\frac{\mathrm{2}\pi}{\mathrm{2}^{\mathrm{10}} }\:=\:\frac{\pi}{\mathrm{2}^{\mathrm{9}} }\:=\:\frac{\pi}{\mathrm{512}} \\ $$

Commented by mathdanisur last updated on 23/Aug/21

Thank you Ser nice

$$\mathrm{Thank}\:\mathrm{you}\:\mathrm{Ser}\:\mathrm{nice} \\ $$

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