Question Number 221100 by Nicholas666 last updated on 24/May/25 $$ \\ $$$$\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\boldsymbol{\mathrm{Prove}}; \\ $$$$\:\:\int_{\mathrm{0}} ^{\:+\infty} \:\frac{\mathrm{4}\centerdot\boldsymbol{\mathrm{cos}}\:\boldsymbol{{x}}\:\centerdot\:\sqrt[{\mathrm{6}\:\:}]{\boldsymbol{\mathrm{sinh}}\:\boldsymbol{{x}}\:}}{\boldsymbol{\mathrm{sinh}}\:\boldsymbol{{x}}\:+\:\boldsymbol{\mathrm{sinh}}\:\mathrm{3}\boldsymbol{{x}}\:+\:\mathrm{4}\:\boldsymbol{\mathrm{sinh}}^{\mathrm{2}} \:\boldsymbol{{x}}\:−\:\mathrm{2}\:\boldsymbol{\mathrm{sinh}}^{\mathrm{2}} \:\mathrm{2}\boldsymbol{{x}}\:+\:\mathrm{4}\:\boldsymbol{\mathrm{sinh}}^{\mathrm{4}} \:\boldsymbol{{x}}\:+\:\mathrm{4}\:\boldsymbol{\mathrm{cosh}}^{\mathrm{4}} \boldsymbol{{x}}}\:\boldsymbol{\mathrm{d}{x}}\:=\:\frac{\boldsymbol{\pi}}{\:\sqrt{\mathrm{6}}\:+\:\mathrm{2}}\:\:\:\:\:\:\: \\ $$$$ \\ $$ Terms…
Question Number 221102 by fantastic last updated on 24/May/25 Answered by mehdee7396 last updated on 25/May/25 $${AB}=\mathrm{2}\sqrt{{ar}}\:\:\&\:\:{BC}=\mathrm{2}\sqrt{{br}}\:\:\:\&\:\:\:{AC}=\mathrm{2}\sqrt{{ab}} \\ $$$$\Rightarrow\sqrt{{ab}}=\left(\sqrt{{a}}+\sqrt{{b}}\right)\sqrt{{r}\:} \\ $$$$\Rightarrow{r}=\frac{{ab}}{{a}+{b}+\mathrm{2}\sqrt{{ab}}}\:\:{or}\:\:\frac{\mathrm{1}}{\:\sqrt{{r}}}=\frac{\mathrm{1}}{\:\sqrt{{a}}}+\frac{\mathrm{1}}{\:\sqrt{{b}}}\: \\ $$$$ \\ $$…
Question Number 221103 by Jgrads last updated on 24/May/25 $$\mathrm{Prove}\::\:\:\:\:\:\forall\mathrm{x}\in\mathrm{IR},\:\forall\mathrm{n}\in\mathrm{IN}^{\ast} \: \\ $$$$\underset{\:\mathrm{0}} {\int}^{\:\frac{\pi}{\mathrm{2}}} \mathrm{ch}\left(\mathrm{2xt}\right)\mathrm{cos}^{\mathrm{2n}} \left(\mathrm{t}\right)\:\mathrm{dt}\:\leqslant\:\mathrm{e}^{\frac{\mathrm{x}^{\mathrm{2}} }{\mathrm{n}}} \underset{\:\mathrm{0}} {\int}^{\:\frac{\pi}{\mathrm{2}}} \mathrm{cos}^{\mathrm{2n}} \left(\mathrm{t}\right)\:\mathrm{dt} \\ $$ Terms of…
Question Number 221099 by Nicholas666 last updated on 24/May/25 $$ \\ $$$$\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\mathrm{Prove}: \\ $$$$\:\:\underset{\:−\pi} {\overset{\:\pi} {\int}}\:\underset{{n}=\mathrm{0}} {\overset{\infty} {\sum}}\:\frac{\mathrm{cos}^{{n}\:+\:\mathrm{1}} \:{x}}{\left({n}\:+\:\mathrm{1}\right)\left(\mathrm{1}\:+\:{e}^{{x}^{\mathrm{2}{n}\:+\mathrm{1}} } \right)}\:\:\mathrm{d}{x}\:=\:\pi\:\mathrm{ln2}\:\:\:\:\: \\ $$$$ \\ $$…
Question Number 221095 by SdC355 last updated on 24/May/25 $$\mathrm{Complex}\:\mathrm{integral} \\ $$$$\mathrm{1}.\:\int_{−\infty} ^{\:+\infty} \:\:\:\frac{\mathrm{d}{z}}{\left({z}^{\mathrm{2}} +\mathrm{1}\right)^{\nu} }=?? \\ $$$$\mathrm{2}.\:\int_{−\infty} ^{+\infty} \:\:\frac{{e}^{\boldsymbol{{i}}\pi{t}} }{{t}^{\mathrm{2}} +\mathrm{1}}\:\mathrm{d}{t}=?? \\ $$$$\mathrm{3}.\:\oint_{\:{C}} \:\frac{\mathrm{1}}{{z}}\:\mathrm{d}{z}=??\:,\:{C};{x}^{\mathrm{2}}…
Question Number 221088 by AkandwanahoDavid last updated on 24/May/25 Terms of Service Privacy Policy Contact: info@tinkutara.com
Question Number 221089 by shunmisaki007 last updated on 24/May/25 $$\mathrm{Find}\: \\ $$$$\left(\mathrm{1}\right)\:\int\frac{\mathrm{1}}{{x}^{\mathrm{9}} }{dx} \\ $$$$\left(\mathrm{2}\right)\:\int\frac{\mathrm{1}}{{x}^{\mathrm{9}} +\mathrm{1}}{dx}. \\ $$ Answered by MrGaster last updated on 24/May/25…
Question Number 221052 by MrGaster last updated on 23/May/25 Answered by Frix last updated on 24/May/25 $$\mathrm{Black}\:\mathrm{needs}\:\mathrm{at}\:\mathrm{least}\:\mathrm{2}\:\mathrm{rounds}\:=\:\mathrm{6}\:\mathrm{draws}. \\ $$$$\mathrm{He}\:\mathrm{wins}\:\mathrm{16}\:\mathrm{out}\:\mathrm{of}\:\mathrm{64}\:\mathrm{different}\:\mathrm{games}\:\mathrm{of} \\ $$$$\mathrm{6}\:\mathrm{draws}\:=\:\mathrm{25\%} \\ $$$$\mathrm{Red}\:\mathrm{needs}\:\mathrm{at}\:\mathrm{least}\:\mathrm{2}\frac{\mathrm{2}}{\mathrm{3}}\:\mathrm{rounds}\:=\:\mathrm{8}\:\mathrm{draws}. \\ $$$$\mathrm{He}\:\mathrm{wins}\:\mathrm{24}\:\mathrm{out}\:\mathrm{of}\:\mathrm{256}\:\mathrm{different}\:\mathrm{games}\:\mathrm{of}…
Question Number 221048 by fantastic last updated on 23/May/25 $$\underset{\mathrm{0}} {\overset{\frac{\pi}{\mathrm{2}}} {\int}}\:\mathrm{cosec}\:\left({x}−\frac{\pi}{\mathrm{3}}\right)\mathrm{cosec}\:\left({x}−\frac{\pi}{\mathrm{6}}\right){dx}\: \\ $$ Answered by vnm last updated on 24/May/25 $$ \\ $$$$\mathrm{the}\:\mathrm{integral}\:\mathrm{diverges},\:\mathrm{but}\:\mathrm{it}'\mathrm{s} \\…
Question Number 221049 by Nicholas666 last updated on 23/May/25 $$ \\ $$$$\:\:\:\:\int\:\:\frac{\mathrm{cos}\:{x}}{\left({x}\:+\:\mathrm{cos}\:{x}\right)^{\mathrm{2}} }\:+\:\frac{\mathrm{2}\left(\mathrm{1}\:−\:\mathrm{sin}\:{x}\right)^{\mathrm{2}} }{\left({x}\:+\:\mathrm{cos}\:{x}\right)^{\mathrm{3}} }\:\:\mathrm{d}{x}\: \\ $$$$\: \\ $$ Terms of Service Privacy Policy Contact:…