Question Number 65290 by mathmax by abdo last updated on 27/Jul/19 $${f}\left({x}\right)\:=\int_{\mathrm{0}} ^{\mathrm{1}} \:\:\frac{{dt}}{{x}+{e}^{{t}} }\:\:\:{with}\:\:\mathrm{0}\leqslant{x}\leqslant\mathrm{1} \\ $$$$\left.\mathrm{1}\right)\:{find}\:{aexplicit}\:{form}\:{of}\:{f}\left({x}\right) \\ $$$$\left.\mathrm{2}\right)\:{calculate}\:\int_{\mathrm{0}} ^{\mathrm{1}} \:\:\frac{{dt}}{\mathrm{2}+{e}^{{t}} } \\ $$$$\left.\mathrm{3}\right)\:{find}\:{g}\left({x}\right)\:=\int_{\mathrm{0}} ^{\mathrm{1}}…
Question Number 65289 by naka3546 last updated on 27/Jul/19 Commented by Prithwish sen last updated on 27/Jul/19 $$\mathrm{let} \\ $$$$\boldsymbol{\Sigma\mathrm{pqr}}=\boldsymbol{\mathrm{A}},\boldsymbol{\Sigma\mathrm{pq}}=\boldsymbol{\mathrm{B}},\boldsymbol{\Sigma\mathrm{p}}=\boldsymbol{\mathrm{C}} \\ $$$$\mathrm{Then}\:\mathrm{from}\:\mathrm{the}\:\mathrm{last}\:\mathrm{three}\:\mathrm{equation}\:\mathrm{we}\:\mathrm{get} \\ $$$$\mathrm{A}+\mathrm{B}+\mathrm{C}=\mathrm{5} \\…
Question Number 65288 by mathmax by abdo last updated on 27/Jul/19 $$\left.\mathrm{1}\right)\:{let}\:{f}\left({x}\right)\:=\int_{\mathrm{0}} ^{+\infty} \:\:\:\frac{{dt}}{{t}^{\mathrm{3}} \:+{x}^{\mathrm{3}} }\:\:\:{with}\:{x}>\mathrm{0} \\ $$$${calculate}\:{f}\left({x}\right) \\ $$$$\left.\mathrm{2}\right)\:{find}\:{also}\:{g}\left({x}\right)\:=\int_{\mathrm{0}} ^{\infty} \:\:\frac{{dt}}{\left({t}^{\mathrm{3}} \:+{x}^{\mathrm{3}} \right)^{\mathrm{2}} }…
Question Number 65287 by mathmax by abdo last updated on 27/Jul/19 $${let}\:{f}\left({x}\right)\:={x}\mid{x}\mid\:\:\:\:\mathrm{2}\pi\:{periodic}\:\:{odd} \\ $$$${developp}\:{f}\:{at}\:{fourier}\:{series} \\ $$ Commented by mathmax by abdo last updated on 28/Jul/19…
Question Number 65286 by mathmax by abdo last updated on 27/Jul/19 $$\left.\mathrm{1}\right){find}\:\:\:{f}\left({a}\right)=\int_{−\infty} ^{+\infty} \:\:{e}^{−{ax}^{\mathrm{2}} } {cos}\left(\mathrm{3}−{x}^{\mathrm{2}} \right){dx}\:\:{with}\:{a}>\mathrm{0} \\ $$$$\left.\mathrm{2}\right)\:{find}\:{the}\:{value}\:{of}\:\int_{\mathrm{0}} ^{\infty} \:{e}^{−\mathrm{3}{x}^{\mathrm{2}} } {cos}\left(\mathrm{3}−{x}^{\mathrm{2}} \right){dx} \\…
Question Number 65285 by mathmax by abdo last updated on 27/Jul/19 $${let}\:{f}\left({x}\right)\:={e}^{−{x}^{\mathrm{2}} } {ln}\left(\mathrm{1}−{x}\right) \\ $$$${developp}\:{f}\:{at}\:{integr}\:{serie}. \\ $$ Commented by mathmax by abdo last updated…
Question Number 130818 by Study last updated on 29/Jan/21 $${if}\:\mid{z}_{\mathrm{1}} \mid=\mid{z}_{\mathrm{2}} \mid=\mid{z}_{\mathrm{3}} \mid=\mathrm{1}\:\: \\ $$$${and}\:\:\:\mid\frac{\mathrm{1}}{{z}_{\mathrm{1}} }\mid+\mid\frac{\mathrm{1}}{{z}_{\mathrm{2}} }\mid+\mid\frac{\mathrm{1}}{{z}_{\mathrm{3}} }\mid=\mathrm{1} \\ $$$$\:{find}\:\mid{z}_{\mathrm{1}} +{z}_{\mathrm{2}} +{z}_{\mathrm{3}} \mid=?\:\:\: \\ $$$${z}_{\mathrm{1}}…
Question Number 130819 by EDWIN88 last updated on 29/Jan/21 $$\:\mathrm{8sin}\:^{\mathrm{3}} \left({x}+\frac{\pi}{\mathrm{6}}\right)=\:\mathrm{cos}\:\left(\mathrm{3}{x}\right) \\ $$$$\:{x}=? \\ $$ Answered by mr W last updated on 29/Jan/21 $${let}\:{u}={x}+\frac{\pi}{\mathrm{6}} \\…
Question Number 65281 by mr W last updated on 27/Jul/19 Commented by mr W last updated on 27/Jul/19 $${segment}\:{of}\:{circle}\:{with}\:{size}\:\mathrm{2}{c}×{d}.\:\left({d}\leqslant{c}\right) \\ $$$${find}\:{the}\:{inscribed}\:{ellipse}\:{with} \\ $$$${maximum}\:{area}. \\ $$…
Question Number 130817 by shaker last updated on 29/Jan/21 Answered by Olaf last updated on 29/Jan/21 $$\mathrm{argth}\left(\mathrm{sh}^{\mathrm{4}} {x}+\mathrm{3sh}^{\mathrm{2}} {x}\right)\:=\:\mathrm{1} \\ $$$$\mathrm{sh}^{\mathrm{4}} {x}+\mathrm{3sh}^{\mathrm{2}} {x}\:=\:\mathrm{th}\left(\mathrm{1}\right)\:=\:\frac{{e}−\frac{\mathrm{1}}{{e}}}{{e}+\frac{\mathrm{1}}{{e}}}\:=\:\frac{{e}^{\mathrm{2}} −\mathrm{1}}{{e}^{\mathrm{2}} +\mathrm{1}}…