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Question Number 90544 by jagoll last updated on 24/Apr/20

∫ (dx/(√(2−cos x)))

$$\int\:\frac{{dx}}{\sqrt{\mathrm{2}−\mathrm{cos}\:{x}}} \\ $$

Answered by $@ty@m123 last updated on 24/Apr/20

∫(dx/(√(2−((1−tan^2 (x/2))/(1+tan^2 (x/2)))))) ∫(dx/(√((2+2tan^2 (x/2)+1−tan^2 (x/2))/(1+tan^2 (x/2))))) ∫(dx/(√((3+tan^2 (x/2))/(sec^2 (x/2))))) ∫((sec^2 (x/2)dx)/(√(3+tan^2 (x/2)))) Let t=tan (x/2) dt=(1/2)sec^2 (x/2)dx I=∫((2dt)/(√(3+t^2 ))) =2ln (t+(√(3+t^2 )))+C

$$\int\frac{{dx}}{\sqrt{\mathrm{2}−\frac{\mathrm{1}−\mathrm{tan}\:^{\mathrm{2}} \frac{{x}}{\mathrm{2}}}{\mathrm{1}+\mathrm{tan}\:^{\mathrm{2}} \frac{{x}}{\mathrm{2}}}}} \\ $$$$\int\frac{{dx}}{\sqrt{\frac{\mathrm{2}+\mathrm{2tan}\:^{\mathrm{2}} \frac{{x}}{\mathrm{2}}+\mathrm{1}−\mathrm{tan}\:^{\mathrm{2}} \frac{{x}}{\mathrm{2}}}{\mathrm{1}+\mathrm{tan}\:^{\mathrm{2}} \frac{{x}}{\mathrm{2}}}}} \\ $$$$\int\frac{{dx}}{\sqrt{\frac{\mathrm{3}+\mathrm{tan}\:^{\mathrm{2}} \frac{{x}}{\mathrm{2}}}{\mathrm{sec}\:^{\mathrm{2}} \frac{{x}}{\mathrm{2}}}}} \\ $$$$\int\frac{\mathrm{sec}\:^{\mathrm{2}} \frac{{x}}{\mathrm{2}}{dx}}{\sqrt{\mathrm{3}+\mathrm{tan}\:^{\mathrm{2}} \frac{{x}}{\mathrm{2}}}} \\ $$$${Let}\:{t}=\mathrm{tan}\:\frac{{x}}{\mathrm{2}} \\ $$$${dt}=\frac{\mathrm{1}}{\mathrm{2}}\mathrm{sec}\:^{\mathrm{2}} \frac{{x}}{\mathrm{2}}{dx} \\ $$$${I}=\int\frac{\mathrm{2}{dt}}{\sqrt{\mathrm{3}+{t}^{\mathrm{2}} }} \\ $$$$=\mathrm{2ln}\:\left({t}+\sqrt{\mathrm{3}+{t}^{\mathrm{2}} }\right)+\mathrm{C} \\ $$

Commented by jagoll last updated on 25/Apr/20

thank you sir

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

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