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Question Number 83603 by jagoll last updated on 04/Mar/20

$$\int \frac{\mathrm{dx}}{1-2\cos \mathrm{x}}$$

Commented by turbo msup by abdo last updated on 04/Mar/20

$$weusethechangementtan\left(\frac{x}{2}\right)=t$$$$\Rightarrow \int \frac{dx}{1-2cosx}=\int \frac{1}{1-2\frac{1-t^2}{1+t^2}}\times \frac{2dt}{1+t^2}$$$$=2\int \frac{dt}{1+t^2-2+2t^2}=2\int \frac{dt}{3t^2-1}$$$$=2\int \frac{dt}{(\sqrt{3}t-1)(\sqrt{3}+1)}$$$$=\int (\frac{1}{\sqrt{3}t-1}-\frac{1}{\sqrt{3}t+1})dt$$$$=\frac{1}{\sqrt{3}}ln\mid \sqrt{3}t-1\mid -\frac{1}{\sqrt{3}}ln\mid \sqrt{3}t+1\mid +c$$$$=\frac{1}{\sqrt{3}}ln\mid \frac{\sqrt{3}t-1}{\sqrt{3}t+1}\mid +c$$$$=\frac{1}{\sqrt{3}}ln\mid \frac{\sqrt{3}tan\left(\frac{x}{2}\right)-1}{\sqrt{3}tan\left(\frac{x}{2}\right)+1}\mid +C$$

Commented by jagoll last updated on 04/Mar/20

$$\mathrm{thank}\mathrm{you}$$

Commented by mathmax by abdo last updated on 04/Mar/20

$$youarewelcome$$

Commented by niroj last updated on 05/Mar/20

$$\int \frac{\mathrm{dx}}{1-2\cos \mathrm{x}}$$$$\mathrm{let},\cos \mathrm{x}=\frac{1-{\tan }^2\frac{\mathrm{x}}{2}}{1+{\tan }^2\frac{\mathrm{x}}{2}}$$$$=\int \frac{1}{1-2\left(\frac{1-{\tan }^2\frac{\mathrm{x}}{2}}{1+{\tan }^2\frac{\mathrm{x}}{2}}\right)}\mathrm{dx}$$$$=\int \frac{(1+{\tan }^2\frac{\mathrm{x}}{2})\mathrm{dx}}{1+{\tan }^2\frac{\mathrm{x}}{2}-2+{2\tan }^2\frac{\mathrm{x}}{2}}$$$$=\int \frac{{\sec }^2\frac{\mathrm{x}}{2}\mathrm{dx}}{{3\tan }^2\frac{\mathrm{x}}{2}-1}$$$$\mathrm{put}\tan \frac{\mathrm{x}}{2}=\mathrm{t}$$$${\sec }^2\frac{\mathrm{x}}{2}\mathrm{dx}=2\mathrm{dt}$$$$=\int \frac{2\mathrm{dt}}{{3\mathrm{t}}^2-1}=\frac{2}{3}\int \frac{1}{{\mathrm{t}}^2-\frac{1}{3}}\mathrm{dt}$$$$=\frac{2}{3}\int \frac{1}{{\left(\mathrm{t}\right)}^2-{\left(\frac{1}{\sqrt{3}}\right)}^2}\mathrm{dt}$$$$=\frac{2}{3}\left[\frac{1}{2.\frac{1}{\sqrt{3}}}\log \frac{\mathrm{t}-\frac{1}{\sqrt{3}}}{\mathrm{t}+\frac{1}{\sqrt{3}}}\right]+\mathrm{C}$$$$=\frac{2.1}{\frac{3.2}{\sqrt{3}}}\left(\log \frac{\mathrm{t}\sqrt{3}-1}{\mathrm{t}\sqrt{3}+1}\right)+\mathrm{C}$$$$=\frac{1}{\sqrt{3}}\left(\log \frac{\sqrt{3}\tan \frac{\mathrm{x}}{2}-1}{\sqrt{3}\tan \frac{\mathrm{x}}{2}+1}\right)+\mathrm{C}$$$$$$

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