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Question Number 77778 by Dah Solu Tion last updated on 10/Jan/20

$$\int \frac{cosx}{2-cosx}dx$$$$$$

Commented by Tony Lin last updated on 10/Jan/20

$$lett=tan\frac{x}{2},\frac{dx}{dt}=\frac{2}{1+t^2}$$$$\int \frac{\frac{1-t^2}{1+t^2}}{\frac{2+2t^2}{1+t^2}-\frac{1-t^2}{1+t^2}}\times \frac{2}{1+t^2}dt$$$$=2\int \frac{1-t^2}{1+3t^2}\times \frac{1}{1+t^2}dt$$$$=2\int \frac{2}{1+3t^2}-\frac{1}{1+t^2}dt$$$$=4\int \frac{1}{1+{\left(\sqrt{3}t\right)}^2}dt-2\int \frac{1}{1+t^2}dt$$$$=\frac{4}{\sqrt{3}}{tan}^{-1}\sqrt{3}t-2{tan}^{-1}t+c$$$$=\frac{4}{\sqrt{3}}{tan}^{-1}\left(\sqrt{3}tan\frac{x}{2}\right)-x+c$$

Commented by john santu last updated on 10/Jan/20

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