Question Number 155619 by mathdanisur last updated on 02/Oct/21 $$\boldsymbol{\Omega}\:=\underset{\:\mathrm{0}} {\overset{\:\mathrm{1}} {\int}}\:\frac{\mathrm{x}^{\mathrm{49}} }{\mathrm{1}\:+\:\mathrm{x}\:+\:\mathrm{x}^{\mathrm{2}} \:+\:\mathrm{x}^{\mathrm{3}} \:…\:\mathrm{x}^{\mathrm{100}} }\:\mathrm{dx}\:=\:? \\ $$ Answered by mindispower last updated on 04/Oct/21…
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Question Number 155618 by mathdanisur last updated on 02/Oct/21 $$\mathrm{if}\:\:\mathrm{x};\mathrm{y}\in\mathbb{R}\:\:\mathrm{and}\:\:\mathrm{x}^{\mathrm{3}} +\mathrm{y}^{\mathrm{3}} =\mathrm{16}\:\:\mathrm{prove}\:\mathrm{that}: \\ $$$$\mathrm{x}^{\mathrm{4}} \:+\:\mathrm{y}^{\mathrm{4}} \:+\:\mathrm{2x}^{\mathrm{2}} \:+\:\mathrm{y}^{\mathrm{2}} \:\geqslant\:\mathrm{4x}\:+\:\mathrm{36} \\ $$ Terms of Service Privacy Policy…
Question Number 24542 by Tinkutara last updated on 20/Nov/17 $${Prove}\:{that}\:{coefficient}\:{of}\:{x}^{{n}} \:{in} \\ $$$$\frac{{a}+{bx}+{cx}^{\mathrm{2}} }{{e}^{{x}} }\:{is}\:\frac{\left(−\mathrm{1}\right)^{{n}} }{{n}!}\left[{cn}^{\mathrm{2}} −\left({b}+{c}\right){n}+{a}\right] \\ $$ Commented by Tinkutara last updated on…
Question Number 90077 by manuel__ last updated on 21/Apr/20 $${lim}_{{x}\rightarrow\infty} \left(\mathrm{sin}\:\left({x}+\frac{\mathrm{1}}{{x}}\right)−{sin}\left({x}\right)\right)=? \\ $$ Commented by jagoll last updated on 21/Apr/20 $$\mathrm{sin}\:\left(\mathrm{x}+\frac{\mathrm{1}}{\mathrm{x}}\right)−\mathrm{sin}\:\left(\mathrm{x}\right)\:=\: \\ $$$$\mathrm{2cos}\:\left(\frac{\mathrm{2x}+\frac{\mathrm{1}}{\mathrm{x}}}{\mathrm{2}}\right)\:\mathrm{sin}\:\left(\frac{\mathrm{1}}{\mathrm{2x}}\right)\:= \\ $$$$\mathrm{2cos}\:\left(\mathrm{x}+\frac{\mathrm{1}}{\mathrm{2x}}\right)\:\mathrm{sin}\:\left(\frac{\mathrm{1}}{\mathrm{2x}}\right)…
Question Number 24540 by Tinkutara last updated on 20/Nov/17 $${Prove}\:{that} \\ $$$$\left({i}\right)\:\underset{{n}=\mathrm{0}} {\overset{\infty} {\sum}}\frac{{n}^{\mathrm{2}} }{{n}!}=\mathrm{2}{e}. \\ $$$$\left({ii}\right)\:\underset{{n}=\mathrm{0}} {\overset{\infty} {\sum}}\frac{{n}^{\mathrm{3}} }{{n}!}=\mathrm{5}{e}. \\ $$$$\left({iii}\right)\:\underset{{n}=\mathrm{0}} {\overset{\infty} {\sum}}\frac{{n}^{\mathrm{4}} }{{n}!}=\mathrm{15}{e}.…
Question Number 24539 by ajfour last updated on 20/Nov/17 Commented by ajfour last updated on 20/Nov/17 $${If}\:{the}\:{system}\:{is}\:{released}\:{with} \\ $$$${the}\:{rod}\:{vertical}\:\left(\theta=\mathrm{0}\right)\:,\:{and}\:{the} \\ $$$${rod}\:{being}\:{attached}\:{to}\:{the}\:{center} \\ $$$${of}\:{disc}\:{with}\:{a}\:{frictionless}\:{axle}, \\ $$$${then}\:{as}\:{the}\:{rod}\:{inclines}\:{down},…
Question Number 90075 by 117427 last updated on 21/Apr/20 $$ \\ $$ Commented by jagoll last updated on 21/Apr/20 $$??\%\sqrt{\left.\%\right]\left\{\right)\left[\mathrm{5}\right.} \\ $$ Terms of Service…
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Question Number 155604 by SANOGO last updated on 02/Oct/21 Answered by Kamel last updated on 02/Oct/21 $$ \\ $$$$\forall{x}\geqslant\mathrm{0}\:\:{x}−\frac{{x}^{\mathrm{3}} }{\mathrm{6}}\leqslant{sin}\left({x}\right)\leqslant{x} \\ $$$$\:\:\:\therefore\:\:\:\underset{{k}=\mathrm{1}} {\overset{{n}} {\sum}}\frac{\mathrm{1}}{{n}+{k}}−\frac{\mathrm{1}}{\mathrm{6}}\underset{{k}=\mathrm{1}} {\overset{{n}}…