Question Number 68220 by ~ À ® @ 237 ~ last updated on 07/Sep/19 $$\:\:\:{Let}\:{consider}\:\left({a}_{{n}} \right)_{{n}} \:{and}\:\left({u}_{{n}} \right)_{{n}} \:{two}\:{reals}\:\:{sequence}\:\: \\ $$$${defined}\:{such}\:{as}\:\:\:{a}_{\mathrm{0}} =\mathrm{1}\:,\:\forall\:{n}>\mathrm{1}\:\:{a}_{{n}+\mathrm{1}} =\underset{{p}=\mathrm{0}} {\overset{{n}} {\sum}}{a}_{{p}}…
Question Number 68219 by ~ À ® @ 237 ~ last updated on 07/Sep/19 $$\:\:\:{Let}\:{consider}\:\left({a}_{{n}} \right)_{{n}} \:{and}\:\left({u}_{{n}} \right)_{{n}} \:{two}\:{reals}\:\:{sequence}\:\: \\ $$$${defined}\:{such}\:{as}\:\:\:{a}_{\mathrm{0}} =\mathrm{1}\:,\:\forall\:{n}>\mathrm{1}\:\:{a}_{{n}+\mathrm{1}} =\underset{{p}=\mathrm{0}} {\overset{{n}} {\sum}}{a}_{{p}}…
Question Number 133725 by Ar Brandon last updated on 23/Feb/21 Commented by Ar Brandon last updated on 23/Feb/21 01:15 AM in India, and 24th February…
Question Number 2645 by Filup last updated on 24/Nov/15 $${A}=\int_{{N}_{\mathrm{1}} } ^{{N}_{\mathrm{2}} } \lfloor{x}\rfloor{dx} \\ $$$$\left({N}_{\mathrm{1}} ,\:{N}_{\mathrm{2}} \right)\in\mathbb{Z},\:\:\:{N}_{\mathrm{1}} <{N}_{\mathrm{2}} \\ $$$$ \\ $$$$\mathrm{Solve}\:\mathrm{for}\:{A} \\ $$…
Question Number 133719 by leena12345 last updated on 23/Feb/21 Commented by leena12345 last updated on 23/Feb/21 $${help} \\ $$ Answered by bramlexs22 last updated on…
Question Number 2629 by Filup last updated on 24/Nov/15 $$\mathrm{Can}\:\mathrm{you}\:\mathrm{evalate}: \\ $$$${A}=\int_{{a}} ^{\:{b}} \lfloor{f}\left({x}\right)\rfloor{dx} \\ $$$$ \\ $$$${for}\:{example}: \\ $$$$\int_{\mathrm{0}.\mathrm{5}} ^{\:\mathrm{2}.\mathrm{5}} \lfloor{x}^{\mathrm{2}} \rfloor{dx} \\ $$…
Question Number 68149 by ~ À ® @ 237 ~ last updated on 06/Sep/19 $$\:{Explicit}\:\:\:{f}\left({a}\right)=\underset{{n}=\mathrm{1}} {\overset{\infty} {\sum}}\:\:\:\frac{\left(−\mathrm{1}\right)^{{n}} }{{n}\left({an}+\mathrm{1}\right)}\:\:\:\: \\ $$ Commented by turbo msup by…
Question Number 68145 by Joel122 last updated on 06/Sep/19 $$\mathrm{Find}\:\mathrm{the}\:\mathrm{arc}\:\mathrm{length},\:\mathrm{given}\:\mathrm{the}\:\mathrm{curve} \\ $$$${x}\left({t}\right)\:=\:\mathrm{sin}\:\left(\pi{t}\right),\:\:{y}\left({t}\right)\:=\:{t}\:,\:\:\mathrm{0}\:\leqslant\:{t}\:\leqslant\:\mathrm{1} \\ $$ Commented by Joel122 last updated on 06/Sep/19 $${x}'\left({t}\right)\:=\:\pi\:\mathrm{cos}\:\left(\pi{t}\right),\:{y}'\left({t}\right)\:=\:\mathrm{1} \\ $$$$ \\…
Question Number 68141 by MJS last updated on 06/Sep/19 $$\mathrm{the}\:\mathrm{2}\:\mathrm{formulas}\:\mathrm{for}\:\mathrm{solving}\:\int\frac{{dx}}{{x}^{\mathrm{3}} +{px}+{q}}\:\mathrm{with} \\ $$$$“\mathrm{nasty}''\:\mathrm{solutions}\:\mathrm{of}\:{x}^{\mathrm{3}} +{px}+{q}=\mathrm{0}\:\mathrm{with}\:{p},\:{q}\:\in\mathbb{R} \\ $$$$ \\ $$$$\mathrm{case}\:\mathrm{1} \\ $$$${D}=\frac{{p}^{\mathrm{3}} }{\mathrm{27}}+\frac{{q}^{\mathrm{2}} }{\mathrm{4}}>\mathrm{0}\:\Rightarrow\:{x}^{\mathrm{3}} +{px}+{q}=\mathrm{0}\:\mathrm{has}\:\mathrm{got}\:\mathrm{1}\:\mathrm{real} \\ $$$$\mathrm{and}\:\mathrm{2}\:\mathrm{conjugated}\:\mathrm{complex}\:\mathrm{solutions}…
Question Number 133674 by liberty last updated on 23/Feb/21 $$\int\:\sqrt{\mathrm{1}+\sqrt{\mathrm{1}+\sqrt{\mathrm{x}}}}\:\mathrm{dx}\:=?\: \\ $$ Answered by EDWIN88 last updated on 23/Feb/21 $$\:\mathrm{let}\:\mathrm{y}\:=\:\sqrt{\mathrm{1}+\sqrt{\mathrm{1}+\sqrt{\mathrm{x}}}}\:\Rightarrow\sqrt{\mathrm{1}+\sqrt{\mathrm{x}}}\:=\:\mathrm{y}^{\mathrm{2}} −\mathrm{1} \\ $$$$\Rightarrow\mathrm{1}+\sqrt{\mathrm{x}}\:=\:\mathrm{y}^{\mathrm{4}} −\mathrm{2y}^{\mathrm{2}} +\mathrm{1}\:,\:\mathrm{x}\:=\:\left(\mathrm{y}^{\mathrm{4}}…