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Question Number 85383 by M±th+et£s last updated on 21/Mar/20

∫(((sin(x))/(cos^(11) (x))))^(1/5)  dx

$$\int\sqrt[{\mathrm{5}}]{\frac{{sin}\left({x}\right)}{{cos}^{\mathrm{11}} \left({x}\right)}}\:{dx} \\ $$

Answered by john santu last updated on 21/Mar/20

∫   (((tan x))^(1/(5 )) /(cos^2 x)) dx = ∫  ((tan x))^(1/(5  ))  d(tan x)  = (5/6) (((tan x)^6 ))^(1/(5  ))  + c

$$\int\:\:\:\frac{\sqrt[{\mathrm{5}\:}]{\mathrm{tan}\:{x}}}{\mathrm{cos}\:^{\mathrm{2}} {x}}\:{dx}\:=\:\int\:\:\sqrt[{\mathrm{5}\:\:}]{\mathrm{tan}\:{x}}\:{d}\left(\mathrm{tan}\:{x}\right) \\ $$$$=\:\frac{\mathrm{5}}{\mathrm{6}}\:\sqrt[{\mathrm{5}\:\:}]{\left(\mathrm{tan}\:{x}\right)^{\mathrm{6}} }\:+\:{c} \\ $$

Commented by M±th+et£s last updated on 21/Mar/20

thank you sir

$${thank}\:{you}\:{sir} \\ $$

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