Question Number 147576 by qaz last updated on 22/Jul/21 $$\left(\mathrm{1}\right)::\:\:\:\:\:\underset{\mathrm{i}=\mathrm{1}} {\overset{\mathrm{n}} {\sum}}\underset{\mathrm{j}=\mathrm{1}} {\overset{\mathrm{n}} {\sum}}\mid\mathrm{i}−\mathrm{j}\mid=? \\ $$$$\left(\mathrm{2}\right)::\:\:\:\:\:\underset{\mathrm{i}=\mathrm{1}} {\overset{\mathrm{n}} {\sum}}\underset{\mathrm{j}=\mathrm{i}} {\overset{\mathrm{n}} {\sum}}\frac{\mathrm{1}}{\mathrm{j}}=? \\ $$$$\left(\mathrm{3}\right)::\:\:\:\:\:\:\underset{\mathrm{i}=\mathrm{1}} {\overset{\mathrm{n}^{\mathrm{2}} } {\sum}}\left[\sqrt{\mathrm{i}}\right]=?…
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Question Number 82020 by M±th+et£s last updated on 17/Feb/20 Commented by mind is power last updated on 17/Feb/20 $${thanx}\:{for}\:{this}\:{beautifull}\:\:{quation} \\ $$ Commented by mind is…
Question Number 81996 by msup trace by abdo last updated on 17/Feb/20 $${calculate}\:{I}_{{n}} =\int\int_{\left[\frac{\mathrm{1}}{{n}},{n}\left[\right.\right.} \:\:{e}^{−{x}^{\mathrm{2}} −\mathrm{3}{y}^{\mathrm{2}} } {dxdy} \\ $$$${and}\:{find}\:{lim}_{{n}\rightarrow+\infty} \:\:{I}_{{n}} \\ $$$${conclude}\:{that}\:\int_{\mathrm{0}} ^{\infty} \:{e}^{−{x}^{\mathrm{2}}…
Question Number 81994 by msup trace by abdo last updated on 17/Feb/20 $${calculate}\:\int\int_{{W}} \left({x}+{y}\right){e}^{{x}−{y}} {dxdy} \\ $$$${with}\:{W}\:{is}\:{the}\:{triangle}\:{limited}\:{by} \\ $$$${o},{A}\left(\mathrm{1},\mathrm{0}\right){and}\:{B}\left(\mathrm{0},\mathrm{1}\right) \\ $$ Terms of Service Privacy…
Question Number 81993 by msup trace by abdo last updated on 17/Feb/20 $${calculate}\:\int\int_{{D}} {ln}\left(\mathrm{1}+{x}+{y}\right){dxdy} \\ $$$${with}\:{D}\:{is}\:{the}\:{triangle}\:{limited}\:{by} \\ $$$${points}\:\mathrm{0},{A}\left(\mathrm{1},\mathrm{0}\right)\:{and}\:{B}\left(\mathrm{0},\mathrm{1}\right) \\ $$ Terms of Service Privacy Policy…
Question Number 147487 by mnjuly1970 last updated on 21/Jul/21 $$ \\ $$$$ \\ $$$$\left({a}\:,\:\mathrm{2}{a}\:+\mathrm{1}\:\right]\cap\left[\:{a}^{\:\mathrm{2}} \:−{a}\:,\:{a}^{\:\mathrm{2}} +\:\mathrm{4}{a}\:+\mathrm{1}\:\right)\neq\:\varnothing \\ $$$$ \\ $$$$\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:{a}\:\in\:? \\ $$$$ \\ $$ Answered…
Question Number 147477 by vvvv last updated on 21/Jul/21 Commented by Rustambek last updated on 21/Jul/21 $${salom} \\ $$ Answered by puissant last updated on…
Question Number 81921 by M±th+et£s last updated on 16/Feb/20 Answered by MJS last updated on 16/Feb/20 $$\int\frac{\sqrt{\sqrt{{x}}+\sqrt{{x}−\mathrm{1}}}}{\:\sqrt{{x}}+\mathrm{1}}{dx}= \\ $$$$\:\:\:\:\:\left[{t}=\sqrt{{x}}+\sqrt{{x}−\mathrm{1}}\:\rightarrow\:{dx}=\frac{{t}^{\mathrm{4}} −\mathrm{1}}{\mathrm{2}{t}^{\mathrm{3}} }{dt}\right] \\ $$$$=\int\frac{\left({t}−\mathrm{1}\right)\left({t}^{\mathrm{2}} +\mathrm{1}\right)}{\:\sqrt{{t}^{\mathrm{3}} }\left({t}+\mathrm{1}\right)}{dt}=…
Question Number 81889 by rajesh4661kumar@gmail.com last updated on 16/Feb/20 Answered by john santu last updated on 16/Feb/20 $${let}\:\sqrt{{x}}\:=\:\mathrm{sin}\:{t}\:\Rightarrow{x}\:=\:\mathrm{sin}\:^{\mathrm{2}} {t} \\ $$$${dx}\:=\:\mathrm{2sin}\:{t}\:\mathrm{cos}\:{t}\:{dt}\: \\ $$$$\Rightarrow{I}\:=\:\int\:\sqrt{\frac{\mathrm{1}−\mathrm{sin}\:{t}}{\mathrm{1}+\mathrm{sin}\:{t}}\:}\:\left(\mathrm{2sin}\:{t}\:\mathrm{cos}\:{t}\:\right)\:{dt} \\ $$$$=\:\int\:\frac{\left(\mathrm{1}−\mathrm{sin}\:{t}\right)\mathrm{2sin}\:{t}\mathrm{cos}\:{t}}{\mathrm{cos}\:{t}}\:{dt}…