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Author: Tinku Tara

prove-that-0-n-x-dx-n-n-1-2-and-0-n-x-dx-n-n-1-2-when-is-floor-and-is-ceil-

Question Number 88852 by M±th+et£s last updated on 13/Apr/20 $${prove}\:{that} \\ $$$$\int_{\mathrm{0}} ^{{n}} \lceil{x}\rceil{dx}=\:\frac{{n}\left({n}+\mathrm{1}\right)}{\mathrm{2}}\:{and}\:\int_{\mathrm{0}} ^{{n}} \lfloor{x}\rfloor{dx}=\frac{{n}\left({n}−\mathrm{1}\right)}{\mathrm{2}} \\ $$$${when}\:\lfloor..\rfloor\:{is}\:{floor}\:{and}\:\lceil..\rceil\:{is}\:{ceil} \\ $$ Answered by mr W last…

sin-3-x-cos-x-dx-

Question Number 23317 by tapan das last updated on 28/Oct/17 $$\int\mathrm{sin}\:^{\mathrm{3}} \mathrm{x}\:\mathrm{cos}\:\mathrm{x}\:\mathrm{dx} \\ $$ Answered by mrW1 last updated on 28/Oct/17 $$=\int\mathrm{sin}^{\mathrm{3}} \:\mathrm{x}\:\mathrm{dsin}\:\mathrm{x} \\ $$$$=\frac{\mathrm{sin}^{\mathrm{4}}…

lim-x-2-x-2-e-x-4e-2-x-2-

Question Number 154390 by liberty last updated on 18/Sep/21 $$\:\underset{{x}\rightarrow\mathrm{2}} {\mathrm{lim}}\frac{{x}^{\mathrm{2}} {e}^{{x}} −\mathrm{4}{e}^{\mathrm{2}} }{{x}−\mathrm{2}}\:? \\ $$ Commented by puissant last updated on 18/Sep/21 $$=\underset{{x}\rightarrow\mathrm{2}} {\mathrm{lim}}\left(\mathrm{2}{xe}^{{x}}…

if-x-1-3-x-36-1-3-3-find-x-1-x-

Question Number 154385 by amin96 last updated on 17/Sep/21 $${if}\:\:\:\sqrt[{\mathrm{3}}]{{x}}−\sqrt[{\mathrm{3}}]{{x}−\mathrm{36}}=\mathrm{3}\:\:\:\:\:\:{find}\:\:\:\:{x}−\frac{\mathrm{1}}{{x}} \\ $$ Answered by amin96 last updated on 18/Sep/21 $${x}={t}^{\mathrm{3}} \:\:\:\Rightarrow\:\:\:{t}−\sqrt[{\mathrm{3}}]{{t}^{\mathrm{3}} −\mathrm{36}}=\mathrm{3}\:\:\Rightarrow\:\:\sqrt[{\mathrm{3}}]{{t}^{\mathrm{3}} −\mathrm{36}}={t}−\mathrm{3} \\ $$$${t}^{\mathrm{3}}…

If-x-y-z-gt-0-then-4x-2-x-y-8y-2-y-z-4z-2-z-x-2x-5y-z-

Question Number 154384 by mathdanisur last updated on 17/Sep/21 $$\mathrm{If}\:\:\mathrm{x};\mathrm{y};\mathrm{z}>\mathrm{0}\:\:\mathrm{then}: \\ $$$$\frac{\mathrm{4x}^{\mathrm{2}} }{\mathrm{x}+\mathrm{y}}\:+\:\frac{\mathrm{8y}^{\mathrm{2}} }{\mathrm{y}+\mathrm{z}}\:+\:\frac{\mathrm{4z}^{\mathrm{2}} }{\mathrm{z}+\mathrm{x}}\:\geqslant\:\mathrm{2x}\:+\:\mathrm{5y}\:+\:\mathrm{z} \\ $$ Answered by ghimisi last updated on 18/Sep/21 $$\mathrm{4}\left(\frac{{x}^{\mathrm{2}}…