Question Number 83603 by jagoll last updated on 04/Mar/20
$$\int\:\frac{\mathrm{dx}}{\mathrm{1}−\mathrm{2cos}\:\mathrm{x}} \\ $$
Commented by turbo msup by abdo last updated on 04/Mar/20
$${we}\:{use}\:{the}\:{changement}\:{tan}\left(\frac{{x}}{\mathrm{2}}\right)={t} \\ $$$$\Rightarrow\int\:\:\frac{{dx}}{\mathrm{1}−\mathrm{2}{cosx}}\:=\int\:\:\frac{\mathrm{1}}{\mathrm{1}−\mathrm{2}\frac{\mathrm{1}−{t}^{\mathrm{2}} }{\mathrm{1}+{t}^{\mathrm{2}} }}×\frac{\mathrm{2}{dt}}{\mathrm{1}+{t}^{\mathrm{2}} } \\ $$$$=\mathrm{2}\int\:\:\:\frac{{dt}}{\mathrm{1}+{t}^{\mathrm{2}} −\mathrm{2}+\mathrm{2}{t}^{\mathrm{2}} }\:=\mathrm{2}\int\:\:\frac{{dt}}{\mathrm{3}{t}^{\mathrm{2}} −\mathrm{1}} \\ $$$$=\mathrm{2}\:\int\:\:\frac{{dt}}{\left(\sqrt{\mathrm{3}}{t}−\mathrm{1}\right)\left(\sqrt{\mathrm{3}}+\mathrm{1}\right)} \\ $$$$=\int\left(\frac{\mathrm{1}}{\sqrt{\mathrm{3}}{t}−\mathrm{1}}−\frac{\mathrm{1}}{\sqrt{\mathrm{3}}{t}\:+\mathrm{1}}\right){dt} \\ $$$$=\frac{\mathrm{1}}{\sqrt{\mathrm{3}}}{ln}\mid\sqrt{\mathrm{3}}{t}−\mathrm{1}\mid−\frac{\mathrm{1}}{\sqrt{\mathrm{3}}}{ln}\mid\sqrt{\mathrm{3}}{t}+\mathrm{1}\mid\:+{c} \\ $$$$=\frac{\mathrm{1}}{\sqrt{\mathrm{3}}}{ln}\mid\frac{\sqrt{\mathrm{3}}{t}−\mathrm{1}}{\sqrt{\mathrm{3}}{t}+\mathrm{1}}\mid\:+{c} \\ $$$$=\frac{\mathrm{1}}{\sqrt{\mathrm{3}}}{ln}\mid\frac{\sqrt{\mathrm{3}}{tan}\left(\frac{{x}}{\mathrm{2}}\right)−\mathrm{1}}{\sqrt{\mathrm{3}}{tan}\left(\frac{{x}}{\mathrm{2}}\right)+\mathrm{1}}\mid\:+{C} \\ $$
Commented by jagoll last updated on 04/Mar/20
$$\mathrm{thank}\:\mathrm{you} \\ $$
Commented by mathmax by abdo last updated on 04/Mar/20
$${you}\:{are}\:{welcome} \\ $$
Commented by niroj last updated on 05/Mar/20
![∫ (( dx)/(1−2cos x)) let , cos x= ((1−tan^2 (x/2))/(1+tan^2 (x/2))) = ∫ (1/(1−2(((1−tan^2 (x/2))/(1+tan^2 (x/2))))))dx = ∫(((1+tan^2 (x/2))dx)/(1+tan^2 (x/2)−2+2tan^2 (x/2))) = ∫((sec^2 (x/2)dx)/(3tan^2 (x/2)−1)) put tan (x/2)= t sec^2 (x/2)dx=2dt = ∫((2dt)/(3t^2 −1))= (2/3)∫(1/(t^2 −(1/3)))dt = (2/3)∫ (( 1)/((t)^2 −((1/(√3)))^2 ))dt = (2/3)[ (1/(2.(1/(√3))))log ((t−(1/(√3)))/(t+(1/(√3))))]+C = ((2.1)/((3.2)/(√3)))( log (( t(√3) −1)/( t(√3) +1)))+C = (1/(√3))(log (( (√3) tan(x/2)−1)/( (√3) tan(x/2)+1)))+C](Q83655.png)
$$\:\:\int\:\frac{\:\:\mathrm{dx}}{\mathrm{1}−\mathrm{2cos}\:\mathrm{x}} \\ $$$$\:\mathrm{let}\:,\:\mathrm{cos}\:\mathrm{x}=\:\frac{\mathrm{1}−\mathrm{tan}^{\mathrm{2}} \frac{\mathrm{x}}{\mathrm{2}}}{\mathrm{1}+\mathrm{tan}^{\mathrm{2}} \frac{\mathrm{x}}{\mathrm{2}}} \\ $$$$\:=\:\:\int\:\:\frac{\mathrm{1}}{\mathrm{1}−\mathrm{2}\left(\frac{\mathrm{1}−\mathrm{tan}^{\mathrm{2}} \frac{\mathrm{x}}{\mathrm{2}}}{\mathrm{1}+\mathrm{tan}^{\mathrm{2}} \frac{\mathrm{x}}{\mathrm{2}}}\right)}\mathrm{dx} \\ $$$$=\:\:\:\:\int\frac{\left(\mathrm{1}+\mathrm{tan}^{\mathrm{2}} \frac{\mathrm{x}}{\mathrm{2}}\right)\mathrm{dx}}{\mathrm{1}+\mathrm{tan}^{\mathrm{2}} \frac{\mathrm{x}}{\mathrm{2}}−\mathrm{2}+\mathrm{2tan}^{\mathrm{2}} \frac{\mathrm{x}}{\mathrm{2}}} \\ $$$$\:=\:\int\frac{\mathrm{sec}^{\mathrm{2}} \frac{\mathrm{x}}{\mathrm{2}}\mathrm{dx}}{\mathrm{3tan}^{\mathrm{2}} \frac{\mathrm{x}}{\mathrm{2}}−\mathrm{1}} \\ $$$$\:\:\mathrm{put}\:\mathrm{tan}\:\frac{\mathrm{x}}{\mathrm{2}}=\:\mathrm{t} \\ $$$$\:\:\:\:\:\:\:\:\mathrm{sec}^{\mathrm{2}} \frac{\mathrm{x}}{\mathrm{2}}\mathrm{dx}=\mathrm{2dt} \\ $$$$\:=\:\:\int\frac{\mathrm{2dt}}{\mathrm{3t}^{\mathrm{2}} −\mathrm{1}}=\:\frac{\mathrm{2}}{\mathrm{3}}\int\frac{\mathrm{1}}{\mathrm{t}^{\mathrm{2}} −\frac{\mathrm{1}}{\mathrm{3}}}\mathrm{dt} \\ $$$$=\:\frac{\mathrm{2}}{\mathrm{3}}\int\:\frac{\:\mathrm{1}}{\left(\mathrm{t}\right)^{\mathrm{2}} −\left(\frac{\mathrm{1}}{\sqrt{\mathrm{3}}}\right)^{\mathrm{2}} }\mathrm{dt} \\ $$$$=\:\frac{\mathrm{2}}{\mathrm{3}}\left[\:\frac{\mathrm{1}}{\mathrm{2}.\frac{\mathrm{1}}{\sqrt{\mathrm{3}}}}\mathrm{log}\:\frac{\mathrm{t}−\frac{\mathrm{1}}{\sqrt{\mathrm{3}}}}{\mathrm{t}+\frac{\mathrm{1}}{\sqrt{\mathrm{3}}}}\right]+\mathrm{C} \\ $$$$=\:\:\frac{\mathrm{2}.\mathrm{1}}{\frac{\mathrm{3}.\mathrm{2}}{\sqrt{\mathrm{3}}}}\left(\:\mathrm{log}\:\frac{\:\mathrm{t}\sqrt{\mathrm{3}}\:\:\:−\mathrm{1}}{\:\mathrm{t}\sqrt{\mathrm{3}}\:+\mathrm{1}}\right)+\mathrm{C} \\ $$$$=\:\:\frac{\mathrm{1}}{\sqrt{\mathrm{3}}}\left(\mathrm{log}\:\frac{\:\:\:\sqrt{\mathrm{3}}\:\mathrm{tan}\frac{\mathrm{x}}{\mathrm{2}}−\mathrm{1}}{\:\:\sqrt{\mathrm{3}}\:\:\:\mathrm{tan}\frac{\mathrm{x}}{\mathrm{2}}+\mathrm{1}}\right)+\mathrm{C} \\ $$$$ \\ $$