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Author: Tinku Tara

The-value-of-C-0-C-1-C-1-C-2-C-n-1-C-n-is-1-n-1-n-n-C-1-C-2-C-n-2-n-1-n-n-C-1-C-2-C-n-3-n-n-n-1-C-1-C-2-C-n-

Question Number 23856 by Tinkutara last updated on 08/Nov/17 $$\mathrm{The}\:\mathrm{value}\:\mathrm{of}\:\left({C}_{\mathrm{0}} \:+\:{C}_{\mathrm{1}} \right)\left({C}_{\mathrm{1}} \:+\:{C}_{\mathrm{2}} \right)…. \\ $$$$\left({C}_{{n}−\mathrm{1}} \:+\:{C}_{{n}} \right)\:\mathrm{is} \\ $$$$\left(\mathrm{1}\right)\:\frac{\left({n}\:+\:\mathrm{1}\right)^{{n}} }{{n}!}\:\centerdot\:{C}_{\mathrm{1}} {C}_{\mathrm{2}} …..{C}_{{n}} \\ $$$$\left(\mathrm{2}\right)\:\frac{\left({n}\:−\:\mathrm{1}\right)^{{n}}…

Question-89384

Question Number 89384 by nimnim last updated on 17/Apr/20 Commented by mathmax by abdo last updated on 17/Apr/20 $${let}\:{try}\:{another}\:{way}\:\:{we}\:{consider}\:{the}\:{diffeomorphism} \\ $$$$\left({u},{v}\right)\rightarrow\left({x},{y}\right)\:/{x}−{y}={u}\:{and}\:{x}+{y}\:={v}\:\Rightarrow{x}\:=\frac{{u}+{v}}{\mathrm{2}}\:{andy}=\frac{−{u}+{v}}{\mathrm{2}} \\ $$$$\Rightarrow\varphi\left({u},{v}\right)=\left(\varphi_{\mathrm{1}} \left({u},{v}\right),\varphi_{\mathrm{2}} \left({u},{v}\right)\right)=\left({x},{y}\right)=\left(\frac{{u}}{\mathrm{2}}+\frac{{v}}{\mathrm{2}},−\frac{{u}}{\mathrm{2}}+\frac{{v}}{\mathrm{2}}\right)…

x-6-1-x-4-1-dx-

Question Number 23849 by Sudipta Jana last updated on 08/Nov/17 $$\int\frac{{x}^{\mathrm{6}} +\mathrm{1}}{{x}^{\mathrm{4}} +\mathrm{1}}{dx} \\ $$ Answered by anunil1234 last updated on 08/Nov/17 $$\int\left(\mathrm{x}^{\mathrm{2}} \:+\frac{\left(−\mathrm{x}^{\mathrm{2}} \:+\mathrm{1}\right)}{\mathrm{x}^{\mathrm{4}}…

0-e-2x-cos-3x-sin-4x-

Question Number 89382 by M±th+et£s last updated on 17/Apr/20 $$\int_{\mathrm{0}} ^{\infty} {e}^{−\mathrm{2}{x}} \:{cos}\left(\mathrm{3}{x}\right)\:{sin}\left(\mathrm{4}{x}\right) \\ $$ Commented by mathmax by abdo last updated on 17/Apr/20 $${cos}\left(\mathrm{3}{x}\right){sin}\left(\mathrm{4}{x}\right)\:={cos}\left(\mathrm{3}{x}\right){cos}\left(\frac{\pi}{\mathrm{2}}−\mathrm{4}{x}\right)…

Question-154910

Question Number 154910 by mr W last updated on 23/Sep/21 Commented by mr W last updated on 23/Sep/21 $${tetrahedron}\:{T}−{ABC}\:{has}\:{edge}\:{lengthes} \\ $$$${a},{b},{c},{u},{v},{w}. \\ $$$${the}\:{faces}\:{of}\:{tetrahedron}\:{T}'−{A}'{B}'{C}' \\ $$$${have}\:{the}\:{distances}\:{d}_{\mathrm{1}}…