Question Number 494 by 123456 last updated on 15/Jan/15 $${a}_{\mathrm{0}} =\mathrm{0} \\ $$$${a}_{\mathrm{1}} =\mathrm{1} \\ $$$${a}_{{n}} =\frac{{a}_{{n}−\mathrm{1}} +{a}_{{n}−\mathrm{2}} +{a}_{{n}−\mathrm{1}} {a}_{{n}−\mathrm{2}} }{\mathrm{3}} \\ $$$$\underset{{n}\rightarrow\infty} {\mathrm{lim}}\:{a}_{{n}} =…
Question Number 492 by 123456 last updated on 25/Jan/15 $$\int\frac{\mathrm{1}−\sqrt{{x}}}{\mathrm{1}+\sqrt{{x}}}{dx} \\ $$ Answered by prakash jain last updated on 14/Jan/15 $${x}={t}^{\mathrm{2}} \\ $$$${dx}=\mathrm{2}{t}\:{dt} \\ $$$$\int\frac{\mathrm{1}−{t}}{\mathrm{1}+{t}}\centerdot\mathrm{2}{t}\:{dt}=\mathrm{2}\int\:\frac{{t}−{t}^{\mathrm{2}}…
Question Number 487 by 13/NaSaNa(N)056565 last updated on 25/Jan/15 $$\int\frac{\mathrm{1}}{\mathrm{1}+\sqrt{\mathrm{2}{x}}}\:\:{dx} \\ $$ Answered by prakash jain last updated on 14/Jan/15 $${x}=\mathrm{2}{t}^{\mathrm{2}} \\ $$$${dx}=\mathrm{4}{t}\:{dt} \\ $$$$\int\frac{\mathrm{1}}{\mathrm{1}+\sqrt{\mathrm{2}{x}}}\:{dx}=\int\:\frac{\mathrm{4}{tdt}}{\mathrm{1}+\mathrm{2}{t}}…
Question Number 131558 by liberty last updated on 06/Feb/21 $$\int_{\mathrm{0}} ^{\mathrm{1}} \frac{\mathrm{dx}}{\:\sqrt{\mathrm{1}−\mathrm{x}^{\mathrm{4}} }}\:=? \\ $$ Answered by Dwaipayan Shikari last updated on 06/Feb/21 $$\int_{\mathrm{0}} ^{\mathrm{1}}…
Question Number 131553 by Salman_Abir last updated on 06/Feb/21 Answered by EDWIN88 last updated on 06/Feb/21 $$ \\ $$$$\mathrm{hahaha}\:….\:\mathrm{du}\:\mathrm{bist}\:\mathrm{groartig} \\ $$ Commented by Dwaipayan Shikari…
Question Number 66018 by Rio Michael last updated on 07/Aug/19 $${prove}\:{by}\:{mathematical}\:{induction}\:{that}\: \\ $$$$\:\underset{{r}=\mathrm{1}} {\overset{{n}} {\sum}}{r}\left({r}\:+\:\mathrm{1}\right)\:=\:\frac{{n}}{\mathrm{3}}\left({n}\:+\:\mathrm{1}\right)\left({n}\:+\:\mathrm{2}\right) \\ $$ Commented by Prithwish sen last updated on 07/Aug/19…
Question Number 483 by 13/NaSaNa(N)056565 last updated on 25/Jan/15 $$\int\frac{\mathrm{1}}{\:\sqrt{{x}−\mathrm{1}}}{dx} \\ $$$$ \\ $$ Answered by 123456 last updated on 12/Jan/15 $$\int\frac{{dx}}{\:\sqrt{{x}−\mathrm{1}}} \\ $$$${u}={x}−\mathrm{1} \\…
Question Number 131552 by liberty last updated on 06/Feb/21 $$\int\:\frac{\mathrm{x}\:\mathrm{dx}}{\left(\mathrm{cot}\:\mathrm{x}+\mathrm{tan}\:\mathrm{x}\right)^{\mathrm{2}} }\:? \\ $$ Answered by EDWIN88 last updated on 06/Feb/21 $$\Leftrightarrow\:\mathrm{cot}\:\mathrm{x}+\mathrm{tan}\:\mathrm{x}\:=\:\frac{\mathrm{1}}{\mathrm{sin}\:\mathrm{x}\:\mathrm{cos}\:\mathrm{x}}\:=\:\frac{\mathrm{2}}{\mathrm{sin}\:\mathrm{2x}} \\ $$$$\Leftrightarrow\:\frac{\mathrm{1}}{\left(\mathrm{cot}\:\mathrm{x}+\mathrm{tan}\:\mathrm{x}\right)^{\mathrm{2}} }=\frac{\mathrm{sin}\:^{\mathrm{2}} \mathrm{2x}}{\mathrm{4}}…
Question Number 66019 by Rio Michael last updated on 07/Aug/19 $$\underset{{x}\rightarrow\infty} {{lim}}\:\left(\mathrm{1}\:+\:\frac{\mathrm{2}}{{x}}\right)^{{x}} \:= \\ $$ Commented by mathmax by abdo last updated on 07/Aug/19 $${let}\:{f}\left({x}\right)=\left(\mathrm{1}+\frac{\mathrm{2}}{{x}}\right)^{{x}}…
Question Number 482 by prakash jain last updated on 12/Jan/15 $$\mathrm{The}\:\mathrm{number}\:\mathrm{1000}!\:\mathrm{has}\:\mathrm{certain}\:\mathrm{number} \\ $$$$\mathrm{of}\:\mathrm{0}{s}\:\mathrm{at}\:\mathrm{the}\:\mathrm{end},\:\mathrm{what}\:\mathrm{the}\:\mathrm{the}\:\mathrm{first}\:\mathrm{non}−\mathrm{zero} \\ $$$$\mathrm{digit}. \\ $$$$\mathrm{1000}!=…\mathrm{d}_{\mathrm{1}} \mathrm{d}_{\mathrm{2}} \mathrm{d}_{\mathrm{3}} \mathrm{D00000}… \\ $$$$\mathrm{where}\:\mathrm{d}_{\mathrm{1}} ,\mathrm{d}_{\mathrm{2}} ,\mathrm{d}_{\mathrm{3}} ,\mathrm{D}\:\mathrm{are}\:\mathrm{digits}.…