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Question Number 162410 by mahdipoor last updated on 29/Dec/21

$$\int \frac{dx}{{(a-cosx)}^2}a>1$$

Answered by mathmax by abdo last updated on 29/Dec/21

$$\mathrm{let}\mathrm{f}\left(\mathrm{a}\right)=\int \frac{\mathrm{dx}}{\mathrm{a}-\mathrm{cosx}}\Rightarrow {\mathrm{f}}^{{}^{\prime }}\left(\mathrm{a}\right)=-\int \frac{\mathrm{dx}}{{(\mathrm{a}-\mathrm{cosx})}^2}\Rightarrow$$ $$\int \frac{\mathrm{dx}}{{(\mathrm{a}-\mathrm{cosx})}^2}=-{\mathrm{f}}^{{}^{\prime }}\left(\mathrm{a}\right)(\mathrm{a}>1)$$ $$\mathrm{f}\left(\mathrm{a}\right){=}_{\tan \left(\frac{\mathrm{x}}{2}\right)=\mathrm{t}}\int \frac{2\mathrm{dt}}{(1+{\mathrm{t}}^2)(\mathrm{a}-\frac{1-{\mathrm{t}}^2}{1+{\mathrm{t}}^2})}$$ $$=2\int \frac{\mathrm{dt}}{\mathrm{a}+{\mathrm{at}}^2-1+{\mathrm{t}}^2}=2\int \frac{\mathrm{dt}}{\mathrm{a}-1+(\mathrm{a}+1){\mathrm{t}}^2}=\frac{2}{\mathrm{a}+1}\int \frac{\mathrm{dt}}{{\mathrm{t}}^2+\frac{\mathrm{a}-1}{\mathrm{a}+1}}$$ $${=}_{\mathrm{t}=\sqrt{\frac{\mathrm{a}-1}{\mathrm{a}+1}}\mathrm{y}}\frac{2}{\mathrm{a}+1}\int \frac{1}{\frac{\mathrm{a}-1}{\mathrm{a}+1}(1+{\mathrm{y}}^2)}\frac{\sqrt{\mathrm{a}-1}}{\sqrt{\mathrm{a}+1}}\mathrm{dy}$$ $$=\frac{2}{\mathrm{a}-1}\times \frac{\sqrt{\mathrm{a}-1}}{\sqrt{\mathrm{a}+1}}\arctan \left(\sqrt{\frac{\mathrm{a}+1}{\mathrm{a}-1}}\mathrm{t}\right)+\mathrm{c}=\frac{2}{\sqrt{{\mathrm{a}}^2-1}}\arctan \left(\sqrt{\frac{\mathrm{a}+1}{\mathrm{a}-1}}\tan \left(\frac{\mathrm{x}}{2}\right)\right)$$ $$\mathrm{f}\left(\mathrm{a}\right)=2{({\mathrm{a}}^2-1)}^{-\frac{1}{2}}\arctan \left(\tan \left(\frac{\mathrm{x}}{2}\right)\mathrm{\Psi }\left(\mathrm{a}\right)\right)$$ $$\mathrm{\Psi }\left(\mathrm{a}\right)=\sqrt{\frac{\mathrm{a}+1}{\mathrm{a}-1}}$$ $$\Rightarrow {\mathrm{f}}^{{}^{\prime }}\left(\mathrm{a}\right)=-\left(2\mathrm{a}\right){({\mathrm{a}}^2-1)}^{-\frac{3}{2}}\arctan (\tan \left(\frac{\mathrm{x}}{2}\right)\mathrm{\Psi }\left(\mathrm{a}\right)$$ $$+2{({\mathrm{a}}^2-1)}^{-\frac{1}{2}}\times \tan \left(\frac{\mathrm{x}}{2}\right)\frac{{\mathrm{\Psi }}^{{}^{\prime }}\left(\mathrm{a}\right)}{1+{\tan }^2\left(\frac{\mathrm{x}}{2}\right){\mathrm{\Psi }}^2\left(\mathrm{a}\right)}$$ $${\mathrm{\Psi }}^{{}^{\prime }}\left(\mathrm{a}\right)=\frac{1}{2}{\left(\frac{\mathrm{a}+1}{\mathrm{a}-1}\right)}^{{}^{\prime }}{\left(\frac{\mathrm{a}+1}{\mathrm{a}-1}\right)}^{-\frac{1}{2}}=\frac{1}{2}{\left(\frac{\mathrm{a}-1+2}{\mathrm{a}-1}\right)}^{{}^{\prime }}{\left(\frac{\mathrm{a}+1}{\mathrm{a}-1}\right)}^{-\frac{1}{2}}$$ $$=-\frac{1}{{(\mathrm{a}-1)}^2\sqrt{\frac{\mathrm{a}+1}{\mathrm{a}-1}}}$$ $$\mathrm{f}\left(\mathrm{a}\right)\mathrm{is}\mathrm{known}...$$

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