Question Number 152273 by peter frank last updated on 27/Aug/21 $$\int\:\frac{\mathrm{tan}\:\theta+\mathrm{tan}\:^{\mathrm{3}} \theta}{\mathrm{1}+\mathrm{tan}\:^{\mathrm{3}} \theta}\mathrm{d}\theta \\ $$ Answered by qaz last updated on 27/Aug/21 $$\int\frac{\mathrm{tan}\:\theta+\mathrm{tan}\:^{\mathrm{3}} \theta}{\mathrm{1}+\mathrm{tan}\:^{\mathrm{3}} \theta}\mathrm{d}\theta…
Question Number 152270 by peter frank last updated on 27/Aug/21 $$\int\frac{\mathrm{5x}+\mathrm{3}}{\:\sqrt{\mathrm{x}^{\mathrm{2}} +\mathrm{4x}+\mathrm{10}}}\mathrm{dx} \\ $$ Answered by Olaf_Thorendsen last updated on 27/Aug/21 $$\mathrm{F}\left({x}\right)\:=\:\int\frac{\mathrm{5}{x}+\mathrm{3}}{\:\sqrt{{x}^{\mathrm{2}} +\mathrm{4}{x}+\mathrm{10}}}\:{dx} \\ $$$$\mathrm{F}\left({u}−\mathrm{2}\right)\:=\:\int\frac{\mathrm{5}{u}−\mathrm{7}}{\:\sqrt{{u}^{\mathrm{2}}…
Question Number 152271 by peter frank last updated on 27/Aug/21 $$\int\left(\mathrm{3x}−\mathrm{2}\right)\sqrt{\mathrm{x}^{\mathrm{2}} +\mathrm{x}+\mathrm{1}}\:\mathrm{dx} \\ $$ Answered by qaz last updated on 27/Aug/21 $$\mathrm{A}=\int\left(\mathrm{3x}−\mathrm{2}\right)\sqrt{\mathrm{x}^{\mathrm{2}} +\mathrm{x}+\mathrm{1}}\mathrm{dx} \\ $$$$=\frac{\mathrm{3}}{\mathrm{2}}\int\left(\mathrm{2x}+\mathrm{1}\right)\sqrt{\mathrm{x}^{\mathrm{2}}…
Question Number 86728 by M±th+et£s last updated on 30/Mar/20 $$\int_{\mathrm{0}} ^{\infty} {ln}\left(\mathrm{1}+\frac{{b}^{\mathrm{2}} }{{x}^{\mathrm{2}} }\right)\:{dx} \\ $$ Commented by mathmax by abdo last updated on 30/Mar/20…
Question Number 86703 by lémùst last updated on 30/Mar/20 $${I}=\int\frac{\mathrm{1}}{{x}^{\mathrm{4}} +\mathrm{1}}{dx} \\ $$ Commented by john santu last updated on 30/Mar/20 $$\mathrm{x}^{\mathrm{4}} +\mathrm{1}\:=\:\left(\mathrm{x}^{\mathrm{2}} −\mathrm{i}\right)\left(\mathrm{x}^{\mathrm{2}} +\mathrm{i}\right)…
Question Number 86675 by M±th+et£s last updated on 30/Mar/20 $$\underset{{x}\rightarrow\infty} {{lim}}\sqrt[{{n}}]{\int_{\mathrm{0}} ^{\mathrm{1}} \left(\mathrm{1}+{x}^{{n}} \right)^{{n}} {dx}}=? \\ $$ Commented by M±th+et£s last updated on 31/Mar/20 $$\underset{{n}\rightarrow\infty}…
Question Number 86671 by M±th+et£s last updated on 30/Mar/20 $$\int{sin}\left({x}\right)\:{arcsin}\left({x}\right) \\ $$ Answered by Rio Michael last updated on 30/Mar/20 $$\mathrm{let}\:\mathrm{me}\:\mathrm{give}\:\mathrm{a}\:\mathrm{try}. \\ $$$$\:\int\:\mathrm{sin}\:\left({x}\right)\:\mathrm{arcsin}\left({x}\right)\:{dx} \\ $$$$\mathrm{using}\:\mathrm{the}\:\mathrm{taylor}\:\mathrm{series}\:\mathrm{expansion}\:\mathrm{for}\:\mathrm{sin}\:\left({x}\right)\:\mathrm{arcsin}\:\left({x}\right)\:\mathrm{centred}\:\mathrm{at}\:\mathrm{0}…
Question Number 152203 by peter frank last updated on 26/Aug/21 $$\mathrm{If}\:\mathrm{x}\:\mathrm{is}\:\mathrm{real}\:\mathrm{show}\:\mathrm{that} \\ $$$$\left(\mathrm{2}+\mathrm{i}\right)^{\left(\mathrm{1}+\mathrm{3i}\right)\mathrm{x}} +\left(\mathrm{2}−\mathrm{i}\right)^{\left(\mathrm{1}−\mathrm{3i}\right)\mathrm{x}} \\ $$$$\mathrm{is}\:\mathrm{also}\:\mathrm{real} \\ $$ Commented by MJS_new last updated on 26/Aug/21…
Question Number 152201 by mnjuly1970 last updated on 26/Aug/21 $$ \\ $$$$\:\:\:…\mathrm{Integral}… \\ $$$$\:\:\:\:\:\:\:\:\:\:\mathrm{I}\::=\:\int_{\mathrm{0}} ^{\:\pi} {ln}\:\left({sin}\left({x}\right)\:\right).{tan}^{\:−\mathrm{1}} \left({cot}\left({x}\right)\right){dx}\overset{?} {=}\:\mathrm{0} \\ $$$$\:\:\:\:\:{proof}\:::\:…. \\ $$$$\:\:\:\:\:\:\mathrm{I}\::=\:\int_{\mathrm{0}} ^{\:\pi} {ln}\:\left({sin}\left({x}\right)\:\right).\:{tan}^{\:−\mathrm{1}} \left(\:{tan}\left(\frac{\pi}{\mathrm{2}}\:−{x}\:\right)\right){dx}…
Question Number 152186 by Tawa11 last updated on 26/Aug/21 $$\int\mathrm{x}^{\mathrm{n}} \:\mathrm{cos}\left(\mathrm{nx}\right)\:\mathrm{dx} \\ $$ Answered by mindispower last updated on 26/Aug/21 $${nx}={y} \\ $$$$\Leftrightarrow\frac{\mathrm{1}}{{n}^{{n}+\mathrm{1}} }\int{y}^{{n}} {cos}\left({x}\right){dx}…