Question Number 47059 by maxmathsup by imad last updated on 04/Nov/18 | ||
$$\:{find}\:\int\:\:\frac{{dx}}{\mathrm{1}+{cos}\:{x}\:+{cos}\left(\mathrm{2}{x}\right)} \\ $$ | ||
Commented by maxmathsup by imad last updated on 06/Nov/18 | ||
$${let}\:{A}\:=\int\:\:\:\frac{{dx}}{\mathrm{1}+{cosx}\:+{cos}\left(\mathrm{2}{x}\right)}\:=\:\int\:\:\frac{{dx}}{\mathrm{1}+{cos}\left({x}\right)+\mathrm{2}{cos}^{\mathrm{2}} {x}−\mathrm{1}} \\ $$$$=\int\:\:\frac{{dx}}{{cosx}\left(\mathrm{1}+\mathrm{2}{cosx}\right)}\:=\int\:\left(\frac{\mathrm{1}}{{cosx}}\:−\frac{\mathrm{2}}{\mathrm{1}+\mathrm{2}{cosx}}\right){dx}\:=\int\:\frac{{dx}}{{cosx}}\:−\int\:\:\frac{\mathrm{2}}{\mathrm{1}+\mathrm{2}{cosx}}{dx} \\ $$$${but}\:\int\:\:\frac{{dx}}{{cosx}}\:=_{{tan}\left(\frac{{x}}{\mathrm{2}}\right)={t}} \:\:\int\:\:\frac{\mathrm{1}}{\frac{\mathrm{1}−{t}^{\mathrm{2}} }{\mathrm{1}+{t}^{\mathrm{2}} }}\:\frac{\mathrm{2}{dt}}{\mathrm{1}+{t}^{\mathrm{2}} }\:=\int\:\:\frac{\mathrm{2}{dt}}{\mathrm{1}−{t}^{\mathrm{2}} }\:=\int\:\left(\frac{\mathrm{1}}{\mathrm{1}−{t}}\:+\frac{\mathrm{1}}{\mathrm{1}+{t}}\right){dt} \\ $$$$={ln}\mid\frac{\mathrm{1}+{t}}{\mathrm{1}−{t}}\mid\:+{c}_{\mathrm{1}} ={ln}\mid\frac{\mathrm{1}+{tan}\left(\frac{{x}}{\mathrm{2}}\right)}{\mathrm{1}−{tan}\left(\frac{{x}}{\mathrm{2}}\right)}\mid={ln}\mid{tan}\left(\frac{{x}}{\mathrm{2}}+\frac{\pi}{\mathrm{4}}\right)\mid\:+{c}_{\mathrm{1}} \:\:{also}\:{we}\:{have} \\ $$$$\int\:\frac{\mathrm{2}}{\mathrm{1}+\mathrm{2}{cosx}}{dt}\:=_{{tan}\left(\frac{{x}}{\mathrm{2}}\right)={t}} \:\:\:\int\:\:\frac{\mathrm{2}}{\mathrm{1}+\mathrm{2}\:\frac{\mathrm{1}−{t}^{\mathrm{2}} }{\mathrm{1}+{t}^{\mathrm{2}} }}\:\frac{\mathrm{2}{dt}}{\mathrm{1}+{t}^{\mathrm{2}} }\:=\mathrm{4}\:\int\:\:\:\:\frac{{dt}}{\mathrm{1}+{t}^{\mathrm{2}} +\mathrm{2}−\mathrm{2}{t}^{\mathrm{2}} } \\ $$$$=\mathrm{4}\:\int\:\:\frac{{dt}}{\mathrm{3}−{t}^{\mathrm{2}} }\:=_{{t}\:=\sqrt{\mathrm{3}}{u}} \:\mathrm{4}\:\:\int\:\:\:\:\:\frac{\sqrt{\mathrm{3}}{du}}{\mathrm{3}\left(\mathrm{1}−{u}^{\mathrm{2}} \right)}\:=\frac{\mathrm{2}\sqrt{\mathrm{3}}}{\mathrm{4}}\:\int\:\:\frac{\mathrm{2}{du}}{\mathrm{1}−{u}^{\mathrm{2}} }\:=\frac{\sqrt{\mathrm{3}}}{\mathrm{2}}{ln}\mid\frac{\mathrm{1}+{u}}{\mathrm{1}−{u}}\mid\:+{c}_{\mathrm{2}} \\ $$$$=\frac{\sqrt{\mathrm{3}}}{\mathrm{2}}{ln}\mid\frac{\mathrm{1}+\frac{{t}}{\sqrt{\mathrm{3}}}}{\mathrm{1}−\frac{{t}}{\sqrt{\mathrm{3}}}}\mid\:+{c}_{\mathrm{2}} =\frac{\sqrt{\mathrm{3}}}{\mathrm{2}}{ln}\mid\frac{\sqrt{\mathrm{3}}+{t}}{\sqrt{\mathrm{3}}−{t}}\mid\:+{c}_{\mathrm{2}} \:\:{finally}\: \\ $$$${A}\:={ln}\mid{tan}\left(\frac{{x}}{\mathrm{2}}+\frac{\pi}{\mathrm{4}}\right)\mid−\frac{\sqrt{\mathrm{3}}}{\mathrm{2}}{ln}\mid\frac{\sqrt{\mathrm{3}}+{tan}\left(\frac{{x}}{\mathrm{2}}\right)}{\sqrt{\mathrm{3}}−{tan}\left(\frac{{x}}{\mathrm{2}}\right)}\mid\:+{C}\:. \\ $$ | ||
Answered by tanmay.chaudhury50@gmail.com last updated on 04/Nov/18 | ||
$$\int\frac{{dx}}{{cosx}+\mathrm{2}{cos}^{\mathrm{2}} {x}} \\ $$$$\int\frac{{dx}}{{cosx}\left(\mathrm{1}+\mathrm{2}{cosx}\right)} \\ $$$$\int\frac{\mathrm{1}+\mathrm{2}{cosx}−\mathrm{2}{cosx}}{{cosx}\left(\mathrm{1}+\mathrm{2}{cosx}\right)}{dx} \\ $$$$\int\frac{{dx}}{{cosx}}−\int\frac{\mathrm{2}{dx}}{\mathrm{2}\left(\frac{\mathrm{1}}{\mathrm{2}}+{cosx}\right)} \\ $$$$\int{secxdx}−\int\frac{{dx}}{\frac{\mathrm{1}}{\mathrm{2}}+\frac{\mathrm{1}−{tan}^{\mathrm{2}} \frac{{x}}{\mathrm{2}}}{\mathrm{1}+{tan}^{\mathrm{2}} \frac{{x}}{\mathrm{2}}}} \\ $$$$\int{secx}\:{dx}−\int\frac{{sec}^{\mathrm{2}} \frac{{x}}{\mathrm{2}}}{\frac{\mathrm{3}}{\mathrm{2}}−\frac{\mathrm{1}}{\mathrm{2}}{tan}^{\mathrm{2}} \frac{{x}}{\mathrm{2}}}{dx} \\ $$$$\int{secxdx}−\mathrm{2}\int\frac{{sec}^{\mathrm{2}} \frac{{x}}{\mathrm{2}}}{\mathrm{3}−{tan}^{\mathrm{2}} \frac{{x}}{\mathrm{2}}}{dx} \\ $$$${I}_{\mathrm{1}} −\mathrm{2}{I}_{\mathrm{2}} \\ $$$${I}_{\mathrm{1}} =\int{secxdx} \\ $$$$={ln}\mid{sex}+{tanx}\mid+{c}_{\mathrm{1}} \\ $$$${I}_{\mathrm{2}} =\int\frac{{sec}^{\mathrm{2}} \frac{{x}}{\mathrm{2}}}{\mathrm{3}−{tan}^{\mathrm{2}} \frac{{x}}{\mathrm{2}}}{dx} \\ $$$${t}={tan}\frac{{x}}{\mathrm{2}}\:\:\:{dt}={sec}^{\mathrm{2}} \frac{{x}}{\mathrm{2}}×\frac{\mathrm{1}}{\mathrm{2}}{dx} \\ $$$${I}_{\mathrm{2}} =\int\frac{\mathrm{2}{dt}}{\left(\sqrt{\mathrm{3}}\:\right)^{\mathrm{2}} −{t}^{\mathrm{2}} } \\ $$$$=\mathrm{2}×\frac{\mathrm{1}}{\mathrm{2}\sqrt{\mathrm{3}}}{ln}\mid\frac{\sqrt{\mathrm{3}}\:+{t}}{\sqrt{\mathrm{3}}\:−{t}}\mid+{c}_{\mathrm{2}} \\ $$$$=\frac{\mathrm{1}}{\sqrt{\mathrm{3}}}{ln}\mid\frac{\sqrt{\mathrm{3}}\:+{tan}\frac{{x}}{\mathrm{2}}}{\sqrt{\mathrm{3}}\:−{tan}\frac{{x}}{\mathrm{2}}}\mid+{c}_{\mathrm{2}} \\ $$$$ \\ $$ | ||
Commented by maxmathsup by imad last updated on 05/Nov/18 | ||
$${thanks}\:{sir}. \\ $$ | ||
Commented by tanmay.chaudhury50@gmail.com last updated on 05/Nov/18 | ||
$${thank}\:{you}\:{sir}... \\ $$ | ||
Answered by ajfour last updated on 04/Nov/18 | ||
$${I}=\int\frac{{dx}}{\mathrm{2cos}\:^{\mathrm{2}} {x}+\mathrm{cos}\:{x}} \\ $$$$\:\:\:{let}\:\:\mathrm{tan}\:\frac{{x}}{\mathrm{2}}\:=\:{t}\:\:\Rightarrow\:\:{dx}\:=\:\frac{\mathrm{2}{dt}}{\mathrm{1}+{t}^{\mathrm{2}} } \\ $$$${I}=\int\frac{\mathrm{2}{dt}}{\left(\mathrm{1}+{t}^{\mathrm{2}} \right)\left[\mathrm{2}\left(\frac{\mathrm{1}−{t}^{\mathrm{2}} }{\mathrm{1}+{t}^{\mathrm{2}} }\right)^{\mathrm{2}} +\frac{\mathrm{1}−{t}^{\mathrm{2}} }{\mathrm{1}+{t}^{\mathrm{2}} }\right]} \\ $$$$\:\:\:=\int\frac{\mathrm{2}\left(\mathrm{1}+{t}^{\mathrm{2}} \right){dt}}{\mathrm{2}\left(\mathrm{1}−{t}^{\mathrm{2}} \right)^{\mathrm{2}} +\mathrm{1}−{t}^{\mathrm{4}} } \\ $$$$\:\:=\:\mathrm{2}\int\frac{\left(\mathrm{1}+{t}^{\mathrm{2}} \right){dt}}{{t}^{\mathrm{4}} −\mathrm{4}{t}^{\mathrm{2}} +\mathrm{3}}\:=\:\mathrm{2}\int\:\frac{\left(\frac{\mathrm{1}}{{t}^{\mathrm{2}} }+\mathrm{1}\right){dt}}{{t}^{\mathrm{2}} +\frac{\mathrm{3}}{{t}^{\mathrm{2}} }−\mathrm{4}} \\ $$$${let}\:{a}\left(\mathrm{1}−\frac{\sqrt{\mathrm{3}}}{{t}^{\mathrm{2}} }\right)+{b}\left(\mathrm{1}+\frac{\sqrt{\mathrm{3}}}{{t}^{\mathrm{2}} }\right)=\frac{\mathrm{1}}{{t}^{\mathrm{2}} }+\mathrm{1} \\ $$$$\Rightarrow\:\:{a}+{b}=\mathrm{1}\:\:\:\&\:\:{b}−{a}\:=\:\frac{\mathrm{1}}{\sqrt{\mathrm{3}}} \\ $$$$\Rightarrow\:\:{a}=\frac{\mathrm{1}}{\mathrm{2}}−\frac{\mathrm{1}}{\mathrm{2}\sqrt{\mathrm{3}}}\:\:;\:\:{b}\:=\:\frac{\mathrm{1}}{\mathrm{2}}+\frac{\mathrm{1}}{\mathrm{2}\sqrt{\mathrm{3}}} \\ $$$$\:\:{I}=\:\left(\mathrm{1}−\frac{\mathrm{1}}{\sqrt{\mathrm{3}}}\right)\int\frac{\left(\mathrm{1}−\frac{\sqrt{\mathrm{3}}}{{t}^{\mathrm{2}} }\right){dt}}{\left({t}+\frac{\sqrt{\mathrm{3}}}{{t}}\right)^{\mathrm{2}} −\mathrm{4}−\mathrm{2}\sqrt{\mathrm{3}}} \\ $$$$\:\:\:\:\:\:\:\:\:+\left(\mathrm{1}+\frac{\mathrm{1}}{\sqrt{\mathrm{3}}}\right)\int\frac{\left(\mathrm{1}+\frac{\sqrt{\mathrm{3}}}{{t}^{\mathrm{2}} }\right){dt}}{\left({t}−\frac{\sqrt{\mathrm{3}}}{{t}}\right)^{\mathrm{2}} −\mathrm{4}+\mathrm{2}\sqrt{\mathrm{3}}} \\ $$$${I}=\:\left(\frac{\sqrt{\mathrm{3}}−\mathrm{1}}{\sqrt{\mathrm{3}}}\right)\frac{\mathrm{1}}{\mathrm{2}\sqrt{\mathrm{4}+\mathrm{2}\sqrt{\mathrm{3}}}}\mathrm{ln}\:\mid\frac{{t}+\frac{\sqrt{\mathrm{3}}}{{t}}−\sqrt{\mathrm{4}+\mathrm{2}\sqrt{\mathrm{3}}}}{{t}+\frac{\sqrt{\mathrm{3}}}{{t}}+\sqrt{\mathrm{4}+\mathrm{2}\sqrt{\mathrm{3}}}}\mid \\ $$$$\:+\left(\frac{\sqrt{\mathrm{3}}+\mathrm{1}}{\sqrt{\mathrm{3}}}\right)\frac{\mathrm{1}}{\mathrm{2}\sqrt{\mathrm{4}−\mathrm{2}\sqrt{\mathrm{3}}}}\mathrm{ln}\:\mid\frac{{t}+\frac{\sqrt{\mathrm{3}}}{{t}}−\sqrt{\mathrm{4}−\mathrm{2}\sqrt{\mathrm{3}}}}{{t}+\frac{\sqrt{\mathrm{3}}}{{t}}+\sqrt{\mathrm{4}−\mathrm{2}\sqrt{\mathrm{3}}}}\mid+{c}\:. \\ $$ | ||
Commented by maxmathsup by imad last updated on 04/Nov/18 | ||
$${thank}\:{you}\:{sir}\:{Ajfour} \\ $$ | ||