Question Number 197099 by Erico last updated on 07/Sep/23 $$\underset{\:\mathrm{0}} {\int}^{\:\frac{\pi}{\mathrm{2}}} \frac{\mathrm{ln}\left(\mathrm{cos}{t}\right)}{\mathrm{sin}{t}}\:\mathrm{d}{t}=??? \\ $$ Answered by witcher3 last updated on 07/Sep/23 $$=\int_{\mathrm{0}} ^{\frac{\pi}{\mathrm{2}}} \frac{\mathrm{sin}\left(\mathrm{t}\right)\mathrm{ln}\left(\mathrm{cos}\left(\mathrm{t}\right)\right)}{\mathrm{1}−\mathrm{cos}^{\mathrm{2}} \left(\mathrm{t}\right)}\mathrm{dt},\mathrm{cos}\left(\mathrm{t}\right)=\mathrm{y}…
Question Number 196690 by MATHEMATICSAM last updated on 29/Aug/23 $$\mathrm{If}\:\left({ax}\right)^{\mathrm{log}{a}} \:=\:\left({bx}\right)^{\mathrm{log}{b}} \:\mathrm{then}\:\mathrm{prove}\:\mathrm{that} \\ $$$${x}\:=\:\frac{\mathrm{1}}{{ab}}\:. \\ $$ Answered by BaliramKumar last updated on 29/Aug/23 $$\mathrm{loga}\centerdot\mathrm{log}\left(\mathrm{ax}\right)\:=\:\mathrm{logb}\centerdot\mathrm{log}\left(\mathrm{bx}\right) \\…
Question Number 196683 by MATHEMATICSAM last updated on 29/Aug/23 $$\mathrm{If}\:{y}\:=\:\frac{{e}^{{x}} \:−\:{e}^{−\:{x}} }{{e}^{{x}} \:+\:{e}^{−\:{x}} }\:\mathrm{then}\:\mathrm{show}\:\mathrm{that} \\ $$$${x}\:=\:\frac{\mathrm{1}}{\mathrm{2}}\mathrm{log}_{{e}} \left(\frac{\mathrm{1}\:+\:{y}}{\mathrm{1}\:−\:{y}}\right). \\ $$ Answered by Frix last updated on…
Question Number 196596 by MATHEMATICSAM last updated on 27/Aug/23 $$\mathrm{If}\:{f}\left({x}\right)\:=\:\mathrm{ln}\left(\frac{\mathrm{1}\:+\:{x}}{\mathrm{1}\:−\:{x}}\right)\:\mathrm{then}\:\mathrm{prove}\:\mathrm{that} \\ $$$${f}\left(\frac{\mathrm{2}{x}}{\mathrm{1}\:+\:{x}^{\mathrm{2}} }\right)\:=\:\mathrm{2}{f}\left({x}\right). \\ $$ Commented by mokys last updated on 27/Aug/23 $${f}\:\left(\frac{\mathrm{2}{x}}{\mathrm{1}+{x}^{\mathrm{2}} }\right)=\:{ln}\left(\frac{\mathrm{1}+\frac{\mathrm{2}{x}}{\mathrm{1}+{x}^{\mathrm{2}} }}{\mathrm{1}−\frac{\mathrm{2}{x}}{\mathrm{1}+{x}^{\mathrm{2}}…
Question Number 196427 by Erico last updated on 24/Aug/23 $$\mathrm{If}\:\:{f}\left({x}\right)=\underset{\:\mathrm{0}} {\int}^{\:{x}} \frac{{dt}}{{t}+{e}^{−{f}\left({t}\right)} },\:\mathrm{determine}\:{f}\left({x}\right) \\ $$ Answered by witcher3 last updated on 25/Aug/23 $$\mathrm{f}\left(\mathrm{0}\right)=\mathrm{0} \\ $$$$\mathrm{f}'\left(\mathrm{x}\right)=\frac{\mathrm{1}}{\mathrm{x}+\mathrm{e}^{−\mathrm{f}\left(\mathrm{x}\right)}…
Question Number 196249 by mathlove last updated on 20/Aug/23 $${log}_{{a}} {x}=\mathrm{30} \\ $$$${log}_{{b}} {x}=\mathrm{70} \\ $$$${log}_{{ab}} {x}=? \\ $$ Answered by AST last updated on…
Question Number 196026 by Erico last updated on 16/Aug/23 $$\underset{\:\mathrm{0}} {\int}^{\:+\infty} \frac{\left({lnt}\right)^{\mathrm{2}} }{\mathrm{1}+{t}^{\mathrm{2}} }{dt} \\ $$ Answered by sniper237 last updated on 16/Aug/23 $$=\:{f}''\left(\mathrm{0}\right)\:\:\:{with}\:\:{f}\left({x}\right)=\int_{\mathrm{0}} ^{\infty}…
Question Number 195895 by Erico last updated on 12/Aug/23 $$\mathrm{Calcul}\:\:\:\:\:\:\:\:\underset{\:\mathrm{0}} {\int}^{\:\frac{\pi}{\mathrm{2}}} \mathrm{t}\sqrt{\mathrm{tan}\left(\mathrm{t}\right)}\:\mathrm{dt} \\ $$ Answered by witcher3 last updated on 13/Aug/23 $$\int_{\mathrm{0}} ^{\infty} \frac{\mathrm{tan}^{−\mathrm{1}} \left(\mathrm{t}\right).\sqrt{\mathrm{t}}}{\mathrm{1}+\mathrm{t}^{\mathrm{2}}…
Question Number 195753 by Erico last updated on 09/Aug/23 $$\mathrm{Calculer}\:\underset{\:\mathrm{0}} {\int}^{\:\mathrm{1}} \frac{{ln}^{\mathrm{2}} {t}}{\:\sqrt{\mathrm{1}−{t}^{\mathrm{2}} }}{dt} \\ $$ Answered by witcher3 last updated on 09/Aug/23 $$\int_{\mathrm{0}} ^{\mathrm{1}}…
Question Number 195278 by Erico last updated on 28/Jul/23 $$\mathrm{calculer}\:\underset{\mathrm{x}\rightarrow\pi} {\mathrm{lim}}\frac{\left(\mathrm{n}+\mathrm{1}\right)\mathrm{sin}\left(\left(\mathrm{n}−\mathrm{1}\right)\mathrm{x}\right)−\left(\mathrm{n}−\mathrm{1}\right)\mathrm{sin}\left(\left(\mathrm{n}+\mathrm{1}\right)\mathrm{x}\right)}{\mathrm{sinx}\:\mathrm{sin}\left(\mathrm{nx}\right)} \\ $$ Terms of Service Privacy Policy Contact: info@tinkutara.com