Question Number 163114 by abdullahhhhh last updated on 03/Jan/22
$$\int\left(\frac{\boldsymbol{\mathrm{cos}}\mathrm{5}\boldsymbol{\mathrm{x}}+\boldsymbol{\mathrm{cos}}\mathrm{4}\boldsymbol{\mathrm{x}}}{\mathrm{1}+\mathrm{2}\boldsymbol{\mathrm{cos}}\mathrm{3}\boldsymbol{\mathrm{x}}}\right)\:\boldsymbol{\mathrm{dx}} \\ $$$$\:\: \\ $$
Answered by blackmamba last updated on 04/Jan/22
$$\:\int\:\frac{\frac{{e}^{\mathrm{5}{x}} +{e}^{−\mathrm{5}{x}} }{\mathrm{2}}\:+\frac{{e}^{\mathrm{4}{x}} +{e}^{−\mathrm{4}{x}} }{\mathrm{2}}}{\mathrm{1}+\mathrm{2}\left(\frac{{e}^{\mathrm{3}{x}} +{e}^{−\mathrm{3}{x}} }{\mathrm{2}}\right)}\:{dx} \\ $$$$\:=\:\int\:\frac{{e}^{\mathrm{5}{x}} +{e}^{−\mathrm{5}{x}} +{e}^{\mathrm{4}{x}} +{e}^{−\mathrm{4}{x}} }{\mathrm{2}+\mathrm{2}{e}^{\mathrm{3}{x}} +\mathrm{2}{e}^{−\mathrm{3}{x}} }\:{dx} \\ $$$$\:{let}\:{e}^{{x}} \:=\:{u}\:\Rightarrow{dx}=\frac{{du}}{{u}} \\ $$$$=\:\int\:\frac{{u}^{\mathrm{5}} +{u}^{−\mathrm{5}} +{u}^{\mathrm{4}} +{u}^{−\mathrm{4}} }{\mathrm{2}+\mathrm{2}{u}^{\mathrm{3}} +\mathrm{2}{u}^{−\mathrm{3}} }\:\frac{{du}}{{u}} \\ $$$$=\:\int\:\frac{{u}^{\mathrm{4}} +{u}^{−\mathrm{6}} +{u}^{\mathrm{3}} +{u}^{−\mathrm{5}} }{\mathrm{2}+\mathrm{2}{u}^{\mathrm{3}} +\mathrm{2}{u}^{−\mathrm{3}} }\:{du} \\ $$$$=\:\int\:\frac{{u}^{\mathrm{10}} +\mathrm{1}+{u}^{\mathrm{9}} +{u}}{\mathrm{2}{u}^{\mathrm{6}} +\mathrm{2}{u}^{\mathrm{9}} +\mathrm{2}{u}^{\mathrm{3}} }\:{du} \\ $$
Commented by mkam last updated on 04/Jan/22
$${false} \\ $$