Question Number 127870 by Eric002 last updated on 02/Jan/21 $$\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\mathrm{2021} \\ $$$${HAPPY}\:{NEW}\:{Year} \\ $$$$\left.\mathrm{1}\right)\int\frac{{x}^{\mathrm{3}} +\mathrm{3}{x}+\mathrm{2}}{\left({x}^{\mathrm{2}} +\mathrm{1}\right)^{\mathrm{2}} \left({x}+\mathrm{1}\right)}{dx} \\ $$$$ \\ $$$$\left.\mathrm{2}\right)\int\frac{\mathrm{2}{cos}\left({x}\right)−{sin}\left({x}\right)}{\mathrm{3}{sin}\left({x}\right)+\mathrm{5}{cos}\left({x}\right)}{dx} \\ $$$$ \\ $$$$\left.\mathrm{3}\right)\int\frac{{tan}\left(\mathrm{2}{x}\right)}{\:\sqrt{{sin}^{\mathrm{6}}…
Question Number 62335 by maxmathsup by imad last updated on 19/Jun/19 $$\left.\mathrm{1}\right)\:{calculate}\:{f}\left({x},{y}\right)\:=\int_{\mathrm{0}} ^{\infty} \:\frac{{e}^{−{xt}} {cos}\left({yt}\right)}{\:\sqrt{{t}}}\:{dt}\:{and}\:{g}\left({x},{y}\right)\:=\int_{\mathrm{0}} ^{\infty} \:\:\frac{{e}^{−{xt}} {sin}\left({yt}\right)}{\:\sqrt{{t}}}\:{dt} \\ $$$${with}\:{x}>\mathrm{0}\:\:{and}\:{y}>\mathrm{0} \\ $$$$\left.\mathrm{2}\right)\:{find}\:{the}\:{values}\:{of}\:\:\int_{\mathrm{0}} ^{\infty} \:\frac{{e}^{−\mathrm{2}{t}} \:{cos}\left({t}\right)}{\:\sqrt{{t}}}\:{dt}\:{and}\:\int_{\mathrm{0}}…
Question Number 62334 by smartsmith459@gmail.com last updated on 19/Jun/19 $${if}\:\alpha^{\mathrm{2}} +\beta^{\mathrm{2}} =\:\left(\alpha+\beta\right)^{\mathrm{2}} −\mathrm{2}\alpha\beta\:{evaluate}\left(\alpha−\beta\right) \\ $$ Answered by Kunal12588 last updated on 20/Jun/19 $$\alpha^{\mathrm{2}} +\beta^{\mathrm{2}} =\left(\alpha+\beta\right)^{\mathrm{2}}…
Question Number 62332 by tanmay last updated on 19/Jun/19 Answered by tanmay last updated on 20/Jun/19 $$\underset{{n}\rightarrow\infty} {\mathrm{lim}}\left[\frac{\sqrt{\mathrm{2}}\:×{n}^{{n}−\frac{\mathrm{1}}{\mathrm{2}}} }{{n}^{{n}+\frac{\mathrm{1}}{\mathrm{2}}} ×\sqrt{\mathrm{2}\pi}\:×{e}^{−{n}} }×\left\{\frac{\left(\mathrm{2}×{n}^{\frac{\mathrm{1}}{{n}}} −\mathrm{1}\right)^{{n}} }{{n}^{\mathrm{2}} }\right\}^{\frac{{n}\left({n}−\frac{\mathrm{1}}{\mathrm{2}}\right)}{{ln}^{\mathrm{2}} {n}}}…
Question Number 62330 by maxmathsup by imad last updated on 19/Jun/19 $${find}\:{the}\:{value}\:{of}\:\int_{\mathrm{0}} ^{\infty} \:\:\frac{{t}^{{a}−\mathrm{1}} }{\left(\mathrm{1}+{t}\right)^{\mathrm{2}} }{dt}\:\:\:{with}\:\:\:\mathrm{0}<{a}<\mathrm{1} \\ $$ Commented by mathmax by abdo last updated…
Question Number 127865 by Ar Brandon last updated on 02/Jan/21 Answered by mathmax by abdo last updated on 03/Jan/21 $$\left.\mathrm{a}\right)\mathrm{wehave}\:\mathrm{u}_{\mathrm{n}+\mathrm{1}\:} =\mathrm{f}\left(\mathrm{u}_{\mathrm{n}} \right)\:\mathrm{with}\:\mathrm{f}\left(\mathrm{x}\right)=\mathrm{x}−\mathrm{x}^{\mathrm{2}} \:\:\mathrm{f}\:\mathrm{is}\:\mathrm{contnue}\: \\ $$$$\mathrm{the}\:\mathrm{fix}\:\mathrm{point}\:\:\mathrm{verify}\:\mathrm{x}=\mathrm{x}−\mathrm{x}^{\mathrm{2}}…
Question Number 127861 by naka3546 last updated on 02/Jan/21 $${How}\:\:{many}\:\:{natural}\:\:{numbers}\:\:{n}\:\leqslant\:\mathrm{2020}\:{such}\:\:{that} \\ $$$$\left({n}\:+\:\mathrm{3}\right)!\:\:{is}\:\:{divisible}\:\:{by}\:\:\mathrm{2}^{{n}} \:? \\ $$ Answered by floor(10²Eta[1]) last updated on 03/Jan/21 $$\mathrm{2}^{\mathrm{n}} \mid\left(\mathrm{n}+\mathrm{3}\right)!\Rightarrow\mathrm{k}\geqslant\mathrm{n} \\…
Question Number 127859 by AgnibhoMukhopadhyay last updated on 02/Jan/21 Answered by liberty last updated on 03/Jan/21 $$\:\frac{\mathrm{3}−\mathrm{5i}}{\mathrm{8}+\mathrm{2i}}\:×\:\frac{\mathrm{8}−\mathrm{2i}}{\mathrm{8}−\mathrm{2i}}\:=\:\frac{\mathrm{24}−\mathrm{46i}−\mathrm{10}}{\mathrm{64}+\mathrm{4}}=\frac{\mathrm{14}−\mathrm{46i}}{\mathrm{68}} \\ $$$$\:=\:\frac{\mathrm{7}}{\mathrm{34}}−\frac{\mathrm{23i}}{\mathrm{34}} \\ $$ Terms of Service Privacy…
Question Number 62322 by aliesam last updated on 19/Jun/19 Commented by arcana last updated on 19/Jun/19 $$\beta_{\mathrm{1}} =\alpha+\alpha^{\mathrm{6}} \\ $$$$\beta_{\mathrm{2}} =\left(\alpha+\alpha^{\mathrm{6}} \right)^{\mathrm{2}} −\mathrm{2}=\alpha^{\mathrm{2}} +\mathrm{2}\alpha^{\mathrm{7}} +\alpha^{\mathrm{12}}…
Question Number 127857 by slahadjb last updated on 02/Jan/21 $$\int\sqrt{{x}}{e}^{{x}} {dx}\:\:? \\ $$ Commented by Dwaipayan Shikari last updated on 02/Jan/21 $$\int{x}^{\frac{\mathrm{1}}{\mathrm{2}}} \underset{{n}=\mathrm{0}} {\overset{\infty} {\sum}}\frac{{x}^{{n}}…