Question Number 131104 by mnjuly1970 last updated on 01/Feb/21 $$\:\:\:\:\:\:\:\:\:\:\:…{advanced}\:\:{calculus}… \\ $$$$\:\:\:\:\:\:\:\int_{\mathrm{0}} ^{\:\infty} \frac{{x}^{{b}−\mathrm{1}} }{\left(\mathrm{1}+{x}^{{a}} \right)^{{s}} }{dx}\overset{???} {=}\frac{\Gamma\left(\frac{{b}}{{a}}\right)\Gamma\left({s}−\frac{{b}}{{a}}\right)}{{a}\Gamma\left({s}\right)} \\ $$ Answered by Ar Brandon last…
Question Number 29 by user1 last updated on 25/Jan/15 $$\mathrm{Differentiate}\:\:\:{e}^{\sqrt{\mathrm{cot}\:{x}}} . \\ $$ Answered by user2 last updated on 03/Nov/14 $$\mathrm{Let}\:\:\:{y}={e}^{\sqrt{\mathrm{cot}\:{x}}} \\ $$$$\mathrm{Put}\:\mathrm{cot}\:{x}\:={t}\:\mathrm{and}\:\sqrt{\mathrm{cot}\:{x}}=\sqrt{{t}}={u},\:\mathrm{so}\:\mathrm{that} \\ $$$$\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:{y}={e}^{{u}}…
Question Number 131103 by mnjuly1970 last updated on 01/Feb/21 $$\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:…{nice}\:\:{calculus}… \\ $$$$\:{prove}\:\:{that}\::: \\ $$$$\:\:\underset{{n}\in\mathbb{N}} {\sum}\left(\frac{\Gamma^{\mathrm{2}} \left({n}\right)}{\mathrm{2}^{−{n}} \left(\mathrm{2}{n}−\mathrm{1}\right)!}\right)=\pi \\ $$$$ \\ $$ Answered by Ar Brandon…
Question Number 78526 by TawaTawa last updated on 18/Jan/20 $$\underset{{x}\rightarrow\mathrm{0}} {\mathrm{lim}}\:\:\left[\frac{\int_{\:\:\mathrm{0}} ^{\:\:\boldsymbol{\mathrm{x}}^{\mathrm{2}} } \:\sqrt{\mathrm{4}\:+\:\boldsymbol{\mathrm{t}}^{\mathrm{3}} }\:\:\boldsymbol{\mathrm{dt}}}{\boldsymbol{\mathrm{x}}^{\mathrm{2}} }\right] \\ $$ Commented by mr W last updated on…
Question Number 12968 by @ANTARES_VY last updated on 08/May/17 $$\left(\boldsymbol{\mathrm{x}}+\mathrm{3}\right)^{\mathrm{2}} +\left(\boldsymbol{\mathrm{y}}−\mathrm{5}\right)^{\mathrm{2}} =\mathrm{45}\:\:\:\boldsymbol{\mathrm{A}}\left(\mathrm{0};\mathrm{11}\right) \\ $$$$\boldsymbol{\mathrm{of}}\:\:\boldsymbol{\mathrm{the}}\:\:\boldsymbol{\mathrm{circle}}\:\:\boldsymbol{\mathrm{to}}\:\:\boldsymbol{\mathrm{the}}\:\:\boldsymbol{\mathrm{point}}\:\:\boldsymbol{\mathrm{of}} \\ $$$$\boldsymbol{\mathrm{trying}}\:\:\boldsymbol{\mathrm{to}}\:\:\boldsymbol{\mathrm{find}}\:\boldsymbol{\mathrm{the}}\:\:\boldsymbol{\mathrm{angular}}\:\:\boldsymbol{\mathrm{coefficient}}. \\ $$ Answered by 433 last updated on 08/May/17…
Question Number 78490 by ~blr237~ last updated on 18/Jan/20 $$\mathrm{Find}\:\mathrm{out}\:\mathrm{A}\:=\underset{\mathrm{n}=\mathrm{2}} {\overset{\infty} {\sum}}\:\:\frac{\zeta\left(\mathrm{n}\right)}{\mathrm{n}\left(−\mathrm{3}\right)^{\mathrm{n}} }\:\:\:\:\:\:\:\:\:\:\:\mathrm{where}\:\:\zeta\left(\mathrm{p}\right)=\underset{\mathrm{n}=\mathrm{1}} {\overset{\infty} {\sum}}\:\frac{\mathrm{1}}{\mathrm{n}^{\mathrm{p}} }\: \\ $$ Answered by mind is power last updated…
Question Number 78489 by ~blr237~ last updated on 18/Jan/20 $$\mathrm{let}\:\:\mathrm{P}\left(\mathrm{x}\right)=\:\mathrm{x}^{\mathrm{5}} −\mathrm{209x}+\mathrm{56}\: \\ $$$$\mathrm{Prove}\:\mathrm{that}\:\mathrm{there}\:\mathrm{exist}\:\mathrm{two}\:\mathrm{roots}\:\:\mathrm{a},\mathrm{b}\:\mathrm{such}\:\mathrm{as}\:\:\:\mathrm{ab}=\mathrm{1} \\ $$$$\mathrm{Find}\:\mathrm{out}\:\mathrm{their}\:\mathrm{sum}\:\left(\:\mathrm{a}+\mathrm{b}=?\right)\:\:\mathrm{and}\:\mathrm{deduce}\:\mathrm{the}\:\mathrm{decomposition}\:\mathrm{of}\:\mathrm{P}\left(\mathrm{x}\right)\:\mathrm{in}\:\mathrm{prime}\:\mathrm{factors}. \\ $$ Answered by MJS last updated on 18/Jan/20 $$\mathrm{2}\:\mathrm{roots}\:\mathrm{with}\:{ab}=\mathrm{1}\:\Rightarrow\:\mathrm{we}\:\mathrm{have}\:\mathrm{a}\:\mathrm{square}\:\mathrm{factor}…
Question Number 143997 by liberty last updated on 20/Jun/21 $$\:{The}\:{maximum}\:{value}\:{of}\: \\ $$$${y}\:=\:\sqrt{\left({x}−\mathrm{3}\right)^{\mathrm{2}} +\left({x}^{\mathrm{2}} −\mathrm{2}\right)^{\mathrm{2}} }−\sqrt{{x}^{\mathrm{2}} +\left({x}^{\mathrm{2}} −\mathrm{1}\right)^{\mathrm{2}} } \\ $$$${is}\:\left({A}\right)\:\sqrt{\mathrm{10}}\:\:\:\:\:\:\:\left({C}\right)\:\mathrm{4}\: \\ $$$$\:\:\:\:\:\:\left({B}\right)\:\mathrm{2}\sqrt{\mathrm{5}}\:\:\:\:\:\left({D}\right)\:\mathrm{10}\: \\ $$ Answered…
Question Number 78400 by ~blr237~ last updated on 17/Jan/20 $$\mathrm{the}\:\mathrm{convolute}\:\mathrm{function}\:\mathrm{of}\:\mathrm{both}\:\mathrm{f}\:\mathrm{and}\:\mathrm{g}\:\mathrm{is}\:\mathrm{marked}\:\mathrm{f}\ast\mathrm{g} \\ $$$$\mathrm{And}\:\mathrm{define}\:\mathrm{by}\:\:\left(\mathrm{f}\ast\mathrm{g}\right)\left(\mathrm{x}\right)=\int_{\mathrm{0}} ^{\mathrm{x}} \mathrm{f}\left(\mathrm{x}−\mathrm{t}\right)\mathrm{g}\left(\mathrm{t}\right)\mathrm{dt} \\ $$$$\mathrm{Let}\:\mathrm{noted}\:\mathrm{E}=\mathrm{the}\:\mathrm{set}\:\mathrm{of}\:\mathrm{function}\:\mathrm{define}\:\mathrm{on}\:\mathbb{R}_{+} \\ $$$$\left.\mathrm{0}\right)\:\mathrm{Prove}\:\mathrm{that}\:\mathrm{there}\:\mathrm{exist}\:\mathrm{a}\:\mathrm{function}\:\mathrm{f}_{\mathrm{0}} \in\mathrm{E}\:\mathrm{such}\:\mathrm{as}\:\mathrm{for}\:\mathrm{all}\:\mathrm{x}>\mathrm{0}\:,\:\int_{\mathrm{0}} ^{\infty} \mathrm{f}_{\mathrm{0}} \left(\mathrm{t}\right)\mathrm{e}^{−\mathrm{xt}} \mathrm{dt}=\mathrm{1} \\ $$$$\left.\mathrm{1}\right)\mathrm{Prove}\:\mathrm{that}\:\left(\mathrm{E},\ast\right)\:\mathrm{is}\:\mathrm{a}\:\mathrm{semigroup}…
Question Number 12814 by syambabu087@gmail.com last updated on 02/May/17 Commented by FilupS last updated on 02/May/17 $${x}\left({y}^{{n}} −{y}\right)+\frac{{dy}}{{dx}}=\mathrm{0} \\ $$$$\frac{{dy}}{{dx}}=−{x}\left({y}^{{n}} −{y}\right) \\ $$$$\frac{\mathrm{1}}{{y}^{{n}} −{y}}\:\frac{{dy}}{{dx}}=−{x} \\…