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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}} \\…