Question Number 140581 by bounhome last updated on 09/May/21 $${prove}\:\:\int{sec}^{\mathrm{2}} {xdx}={tanx} \\ $$ Answered by MJS_new last updated on 09/May/21 $$\mathrm{tan}\:{x}\:=\frac{\mathrm{sin}\:{x}}{\mathrm{cos}\:{x}} \\ $$$$\frac{{d}}{{dx}}\left[\frac{{u}\left({x}\right)}{{v}\left({x}\right)}\right]=\frac{{u}'\left({x}\right){v}\left({x}\right)−{u}\left({x}\right){v}'\left({x}\right)}{\left({v}\left({x}\right)\right)^{\mathrm{2}} } \\…
Question Number 140576 by SOMEDAVONG last updated on 09/May/21 $$\mathrm{Find}\:\mathrm{the}\:\mathrm{sum}\:\mathrm{of}\underset{\mathrm{n}=\mathrm{1}} {\overset{+\propto} {\sum}}\frac{\mathrm{2}^{\mathrm{n}} }{\mathrm{n}^{\mathrm{3}} }=? \\ $$ Answered by Dwaipayan Shikari last updated on 09/May/21 $${Li}_{\mathrm{3}}…
Question Number 75033 by naka3546 last updated on 06/Dec/19 $$\frac{\mathrm{4}}{\mathrm{11}}\:<\:\frac{{x}}{{y}}\:<\:\frac{\mathrm{3}}{\mathrm{8}} \\ $$$${x},\:{y}\:\:\in\:\:\mathbb{Z}^{+} \\ $$$${min}\:\left\{{x}+{y}\right\}\:\:=\:\:? \\ $$ Answered by mr W last updated on 06/Dec/19 $$\frac{\mathrm{4}}{\mathrm{11}}<\frac{{x}}{{y}}<\frac{\mathrm{3}}{\mathrm{8}}…
Question Number 140534 by SOMEDAVONG last updated on 09/May/21 Answered by EDWIN88 last updated on 09/May/21 $$\left(\mathrm{i}\right)\:=\:\frac{\mathrm{16x}−\mathrm{24}}{\left(\mathrm{x}−\mathrm{1}\right)\left(\mathrm{x}−\mathrm{3}\right)\left(\mathrm{x}+\mathrm{3}\right)}\:=\:\frac{\mathrm{a}}{\mathrm{x}−\mathrm{1}}+\frac{\mathrm{b}}{\mathrm{x}−\mathrm{3}}+\frac{\mathrm{c}}{\mathrm{x}+\mathrm{3}} \\ $$$$\mathrm{a}\:=\:\left[\frac{\mathrm{16x}−\mathrm{24}}{\left(\mathrm{x}−\mathrm{3}\right)\left(\mathrm{x}+\mathrm{3}\right)}\:\right]_{\mathrm{x}=\mathrm{1}} =\:\frac{−\mathrm{8}}{−\mathrm{8}}\:=\mathrm{1} \\ $$$$\mathrm{b}=\:\left[\frac{\mathrm{16x}−\mathrm{24}}{\left(\mathrm{x}−\mathrm{1}\right)\left(\mathrm{x}+\mathrm{3}\right)}\:\right]_{\mathrm{x}=\mathrm{3}} =\:\frac{\mathrm{48}−\mathrm{24}}{\mathrm{6}.\mathrm{2}}=\mathrm{2} \\ $$$$\mathrm{c}\:=\:\left[\frac{\mathrm{16x}−\mathrm{24}}{\left(\mathrm{x}−\mathrm{1}\right)\left(\mathrm{x}−\mathrm{3}\right)}\:\right]_{\mathrm{x}=−\mathrm{3}}…
Question Number 140529 by SOMEDAVONG last updated on 09/May/21 Answered by Rasheed.Sindhi last updated on 09/May/21 $$\frac{\mathrm{1}}{{x}^{\mathrm{2}} \left({x}+\mathrm{2}\right)}−\frac{\mathrm{5}}{\left({x}−\mathrm{2}\right)\left({x}+\mathrm{2}\right)}−\frac{\mathrm{4}}{{x}−\mathrm{2}} \\ $$$$=\frac{\mathrm{1}\left({x}−\mathrm{2}\right)−\mathrm{5}\left({x}^{\mathrm{2}} \right)−\mathrm{4}\left(\:{x}^{\mathrm{2}} \left({x}+\mathrm{2}\right)\:\right)}{{x}^{\mathrm{2}} \left({x}−\mathrm{2}\right)\left({x}+\mathrm{2}\right)} \\ $$$$=\frac{{x}−\mathrm{2}−\mathrm{5}{x}^{\mathrm{2}}…
Question Number 9458 by geovane10math last updated on 09/Dec/16 $$\mathrm{If}\:\mathrm{the}\:\mathrm{zeta}\:\mathrm{function}\:\mathrm{of}\:\mathrm{2}\:\mathrm{is} \\ $$$$\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\boldsymbol{\zeta}\left(\mathrm{2}\right)\:=\:\underset{\boldsymbol{{n}}=\mathrm{1}} {\overset{\infty} {\sum}}\:\frac{\mathrm{1}}{\boldsymbol{{n}}^{\mathrm{2}} } \\ $$$$\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\boldsymbol{\zeta}\left(\mathrm{2}\right)\:=\:\frac{\boldsymbol{\pi}^{\mathrm{2}} }{\mathrm{6}} \\ $$$$\mathrm{the}\:\mathrm{sum}\:\mathrm{of}\:\mathrm{infinite}\:\boldsymbol{\mathrm{rational}}\:\mathrm{numbers}, \\ $$$$\mathrm{why}\:\mathrm{converges}\:\mathrm{for}\:\frac{\boldsymbol{\pi}^{\mathrm{2}} }{\mathrm{6}},\:\mathrm{an}\:\boldsymbol{\mathrm{irrational}} \\ $$$$\mathrm{number}?…
Question Number 140513 by mathocean1 last updated on 08/May/21 $$\mathrm{n}\:\in\:\mathbb{N}^{\ast} \:\mathrm{and}\:\mathrm{k}\:\in\:\mathbb{N}^{\ast} . \\ $$$$\mathrm{Given}\:\mathrm{0}\leqslant\mathrm{k}\leqslant\mathrm{n}−\mathrm{1}. \\ $$$$\mathrm{Show}\:\mathrm{that}\: \\ $$$$\frac{\mathrm{1}}{\mathrm{n}}\mathrm{ln}\left(\mathrm{1}+\frac{\mathrm{k}}{\mathrm{n}}\right)\leqslant\underset{\mathrm{1}+\frac{\mathrm{k}}{\mathrm{n}}} {\overset{\mathrm{1}+\frac{\mathrm{k}+\mathrm{1}}{\mathrm{n}}} {\int}}\mathrm{lnx}\:\mathrm{dn}\leqslant\frac{\mathrm{1}}{\mathrm{n}}\mathrm{ln}\left(\mathrm{1}+\frac{\mathrm{k}+\mathrm{1}}{\mathrm{n}}\right) \\ $$ Terms of Service…
Question Number 140503 by ERA last updated on 08/May/21 $$\boldsymbol{\mathrm{P}}\mathrm{rove}\:\mathrm{it} \\ $$$$\sqrt{\boldsymbol{{a}}+\sqrt{\boldsymbol{{b}}}}=\sqrt{\frac{\boldsymbol{{a}}+\sqrt{\boldsymbol{{a}}^{\mathrm{2}} +\boldsymbol{{b}}}}{\mathrm{2}}+}\sqrt{\frac{\boldsymbol{{a}}−\sqrt{\boldsymbol{{a}}^{\mathrm{2}} −\boldsymbol{{b}}}}{\mathrm{2}}} \\ $$ Commented by MJS_new last updated on 08/May/21 $$\mathrm{it}'\mathrm{s}\:\mathrm{wrong} \\…
Question Number 74957 by vishalbhardwaj last updated on 04/Dec/19 $$\mathrm{what}\:\mathrm{is}\:\mathrm{the}\:\mathrm{exact}\:\mathrm{difference}\:\mathrm{between} \\ $$$$\mathrm{relation}\:\mathrm{and}\:\mathrm{function}\:?? \\ $$ Terms of Service Privacy Policy Contact: info@tinkutara.com
Question Number 9404 by alfat123 last updated on 05/Dec/16 Answered by geovane10math last updated on 06/Dec/16 $$\mathrm{I}\:\mathrm{can}'\mathrm{t}\:\mathrm{read}… \\ $$$${f}'\left({x}\right)\:=\:\mathrm{2}{x}\:+\:\mathrm{2} \\ $$$$\int{f}'\left({x}\right)\:{dx}\:=\:\int\:\mathrm{2}{x}\:+\:\mathrm{2}\:{dx}\:=\:\int\mathrm{2}{x}\:{dx}\:+\:\int\mathrm{2}\:{dx} \\ $$$$=\:{x}^{\mathrm{2}} \:+\:\mathrm{2}{x}\:+\:{C} \\…