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Question-147873

Question Number 147873 by 0731619 last updated on 24/Jul/21 Answered by Olaf_Thorendsen last updated on 24/Jul/21 $$\mathrm{S}\:=\:\underset{{n}=\mathrm{0}} {\overset{\infty} {\sum}}\left(\frac{\mathrm{1}}{\mathrm{3}{n}+\mathrm{2}}−\frac{\mathrm{1}}{\mathrm{3}{n}+\mathrm{4}}\right) \\ $$$$\mathrm{S}\:=\:\underset{{n}=\mathrm{0}} {\overset{\infty} {\sum}}\frac{\mathrm{2}}{\left(\mathrm{3}{n}+\mathrm{2}\right)\left(\mathrm{3}{n}+\mathrm{4}\right)} \\ $$$$\mathrm{S}\:=\:\frac{\mathrm{2}}{\mathrm{9}}\underset{{n}=\mathrm{0}}…

Question-147879

Question Number 147879 by 0731619 last updated on 24/Jul/21 Answered by puissant last updated on 24/Jul/21 $$\mathrm{I}=\int\frac{\mathrm{1}}{\mathrm{x}^{\mathrm{5}} +\mathrm{1}}\mathrm{dx} \\ $$$$=\int\frac{\mathrm{dx}}{\left(\mathrm{x}+\mathrm{1}\right)\left(\mathrm{x}^{\mathrm{4}} −\mathrm{x}^{\mathrm{3}} +\mathrm{x}^{\mathrm{2}} −\mathrm{x}+\mathrm{1}\right)} \\ $$$$=\int\frac{\frac{\mathrm{1}}{\mathrm{5}}}{\mathrm{x}+\mathrm{1}}\mathrm{dx}+\int\frac{−\frac{\mathrm{1}}{\mathrm{5}}\mathrm{x}^{\mathrm{3}}…

secx-dx-

Question Number 147867 by Khalmohmmad last updated on 24/Jul/21 $$\int\mathrm{sec}{x}\:{dx}=? \\ $$ Answered by Olaf_Thorendsen last updated on 24/Jul/21 $$\mathrm{F}\left({x}\right)\:=\:\int\mathrm{sec}{x}\:{dx} \\ $$$$\mathrm{F}\left({x}\right)\:=\:\int\frac{{dx}}{\mathrm{cos}{x}} \\ $$$$\mathrm{Let}\:{t}\:=\:\mathrm{tan}\frac{{x}}{\mathrm{2}} \\…

Question-147800

Question Number 147800 by 0731619 last updated on 23/Jul/21 Answered by Olaf_Thorendsen last updated on 23/Jul/21 $$\mathbb{R}\:=\:\underset{{n}\in\mathbb{Z}} {\cup}\mathrm{E}_{{n}} ,\:\mathrm{E}_{{n}} \:=\:\left[{n},{n}+\mathrm{1}\left[\right.\right. \\ $$$$\forall{x}\in\mathrm{E}_{{n}} ,\:\left\{{x}\right\}\:=\:{x}−\lfloor{x}\rfloor\:=\:{x}−{n} \\ $$$$\forall{x}\in\mathrm{E}_{{n}}…

Question-147784

Question Number 147784 by Khalmohmmad last updated on 23/Jul/21 Answered by Olaf_Thorendsen last updated on 23/Jul/21 $${F}\left({x}\right)\:=\:\int\frac{{dx}}{\:\sqrt{\mathrm{7}{x}^{\mathrm{2}} −\mathrm{8}}} \\ $$$${F}\left({x}\right)\:=\:\frac{\mathrm{1}}{\:\sqrt{\mathrm{7}}}\int\frac{{dx}}{\:\sqrt{{x}^{\mathrm{2}} −\frac{\mathrm{8}}{\mathrm{7}}}} \\ $$$${F}\left({x}\right)\:=\:\frac{\mathrm{1}}{\:\sqrt{\mathrm{7}}}\mathrm{argch}\left(\frac{{x}}{\:\sqrt{\frac{\mathrm{8}}{\:\mathrm{7}}}}\right)+\mathrm{C} \\ $$$${F}\left({x}\right)\:=\:\frac{\mathrm{1}}{\:\sqrt{\mathrm{7}}}\mathrm{argch}\left(\frac{\mathrm{1}}{\mathrm{2}}\sqrt{\frac{\mathrm{7}}{\mathrm{2}}}{x}\right)+\mathrm{C}…

Question-147768

Question Number 147768 by aliibrahim1 last updated on 23/Jul/21 Answered by liberty last updated on 23/Jul/21 $${Let}\:{BC}\:=\:{x}\:\Rightarrow\mathrm{tan}\:\mathrm{44}°=\frac{{CD}}{{x}} \\ $$$$\Rightarrow{CD}={x}\:\mathrm{tan}\:\mathrm{44}° \\ $$$$\mathrm{L}{et}\:{AC}\:={y}\Rightarrow\mathrm{tan}\:\mathrm{48}°=\frac{{CD}}{{y}} \\ $$$$\Rightarrow{CD}={y}\:\mathrm{tan}\:\mathrm{48}° \\ $$$${where}\:{y}=\sqrt{\mathrm{30}^{\mathrm{2}}…