Question Number 216918 by mathocean1 last updated on 24/Feb/25 $${give}\:{a}\:{recurrence}\:{relation}\:{for}\:{I}_{{n}} . \\ $$$${I}_{{n}} =\int_{\mathrm{0}} ^{\mathrm{1}} \frac{{x}^{{n}} }{{x}+\mathrm{3}}{dx},\:\forall{n}\:\in\:\mathbb{N}. \\ $$ Answered by Wuji last updated on…
Question Number 216913 by Bhutto last updated on 24/Feb/25 $$\mathrm{6}\frac{\mathrm{1}}{\mathrm{4}}\%\: \\ $$ Answered by MrGaster last updated on 24/Feb/25 $$\mathrm{6}\frac{\mathrm{1}}{\mathrm{4}}\%=\frac{\mathrm{25}}{\mathrm{4}}\%=\frac{\mathrm{25}}{\mathrm{400}}=\frac{\mathrm{1}}{\mathrm{16}} \\ $$ Terms of Service…
Question Number 216910 by Marzuk last updated on 24/Feb/25 $${Ed}:.\mathrm{06} \\ $$$${a}\:{function}\:\delta\left({x}\right)\:{is}\:{a}\:{composite}\:{function} \\ $$$${which}\:{is}\:{as}\:{follow}\: \\ $$$$\left[\left\{\left({f}\:\circ\:{g}\right)\left({x}\right)\right\}\:\circ\:\left\{\left({g}\:\circ\:{f}\right)\left({x}\right)\right\}\right]\:\circ\:\left[\left\{\left({f}\:'\:\circ\:{g}\:'\right)\left({x}\right)\right\}\:\circ\:\left\{\left({g}\:'\:\circ\:{f}\:'\right)\left({x}\right)\right\}\right] \\ $$$${where} \\ $$$${f}\left({x}\right)\:=\:\underset{{n}\:=\:\mathrm{1}} {\overset{\infty} {\prod}}\:\frac{{nx}^{\mathrm{3}} \:−\:{nx}^{\mathrm{2}} \:−\:{nx}\:\:−{n}}{{n}^{\mathrm{3}} {x}\:−\:{n}^{\mathrm{2}}…
Question Number 216907 by MrGaster last updated on 24/Feb/25 $$\mathrm{Prove}:{n}!=\mathrm{1}+\underset{{k}=\mathrm{1}} {\overset{\infty} {\sum}}\frac{{k}^{{n}} }{{e}^{{k}} }−\underset{{k}=\mathrm{1}} {\overset{\infty} {\sum}}\frac{{B}_{{k}} \mathrm{sin}\left(\pi{k}\right)\left({n}−{k}\right)!}{\pi{k}} \\ $$ Answered by MrGaster last updated on…
Question Number 216906 by MrGaster last updated on 24/Feb/25 $$\mathrm{Prove}:\Gamma\left({x}\right)=\frac{{x}^{{x}} }{\left(\mathrm{2}\pi\right)^{{x}−\mathrm{1}} }\underset{{k}=\mathrm{1}} {\overset{\infty} {\prod}}\frac{{k}^{\mathrm{2}\left({x}−\mathrm{1}\right)} }{\underset{{i}=\mathrm{1}} {\overset{{x}−\mathrm{1}} {\prod}}\left[{k}^{\mathrm{2}} −\left(\frac{{i}}{\mathrm{2}}\right)^{\mathrm{2}} \right]} \\ $$ Terms of Service Privacy…
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Question Number 216830 by MrGaster last updated on 22/Feb/25 $$\mathrm{Prove}:\forall{x}\in\mathbb{R},\mid\mathrm{cos}\:{x}\mid+\mid\mathrm{cos}\:\mathrm{2}{x}\mid+\ldots+\mid\mathrm{cos}\:{nx}\mid\geq\frac{{n}−\mathrm{1}}{\mathrm{2}}\left({n}\in\mathbb{Z}_{>\mathrm{0}} \right)\:\: \\ $$ Commented by MathematicalUser2357 last updated on 25/Feb/25 $$\mathrm{Or}\:\mathrm{you}\:\mathrm{could}\:\mathrm{do}\:\left\{{n}\mid{n}\in\mathbb{Z}\wedge{n}>\mathrm{0}\right\}\:\left(\mathrm{or}\:\left\{{n}\mid{n}\in\mathbb{N}\right\}\right) \\ $$ Answered by…
Question Number 216820 by Ismoiljon_008 last updated on 22/Feb/25 $$ \\ $$$$\:\:\:{f}\left({x}\right)\:=\:{ax}^{\mathrm{4}} \:+\:{bx}^{\mathrm{3}} \:+\:{cx}^{\mathrm{2}} \:+\:{dx}\:+\:{e} \\ $$$$\:\:\:{f}\left(\mathrm{1}\right)\:=\:\mathrm{2} \\ $$$$\:\:\:{f}\left(\mathrm{2}\right)\:=\:\mathrm{3} \\ $$$$\:\:\:{f}\left(\mathrm{3}\right)\:=\:\mathrm{4} \\ $$$$\:\:\:{f}\left(\mathrm{4}\right)\:=\:\mathrm{5}\: \\ $$$$\:\:\:{f}\left(\mathrm{0}\right)\:=\:\mathrm{25}…
Question Number 216807 by GDVilla last updated on 21/Feb/25 $$\mathrm{Uh}\:\mathrm{guys}\:\mathrm{is}\:\mathrm{the}\:\mathrm{speed}\:\mathrm{formula} \\ $$$$\frac{{d}}{{t}} \\ $$$$\mathrm{or} \\ $$$$\mathrm{li}\underset{\Delta{t}\rightarrow\mathrm{0}} {\mathrm{m}}\frac{\Delta{d}}{\Delta{t}} \\ $$ Answered by MrGaster last updated on…
Question Number 216722 by issac last updated on 17/Feb/25 $$\:\:\:\:\:\:\: \\ $$ Answered by MrGaster last updated on 17/Feb/25 $$\int_{\mathrm{0}} ^{\infty} \left(\frac{\mathrm{1}}{\ell}−\frac{\mathrm{1}}{{p}}\right){d}\ell=\int_{\mathrm{0}} ^{\infty} \left(\frac{{p}−\ell}{\ell{p}}\right){d}\ell \\…