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Question Number 98214 Answers: 2 Comments: 2
Question Number 98208 Answers: 0 Comments: 2
$$\mathrm{suppose}\:\mathrm{a}\:\mathrm{force}\:\mathrm{given}\:\mathrm{as}\:{F}_{\mathrm{1}} \:=\:\mathrm{24}\:{N}\:\mathrm{and}\:{F}_{\mathrm{2}} \:=\:\mathrm{50}\:{N}\:\mathrm{act}\:\mathrm{through}\: \\ $$$$\mathrm{points}\:{AB}\:\mathrm{and}\:{AC}\:\mathrm{where}\:\:{OA}\:=\:\mathrm{2}{i}\:+\mathrm{3}{j}\:,\:{OB}\:=\:\mathrm{5}{i}\:+\:\mathrm{6}{j}\:\:\mathrm{and}\: \\ $$$${OC}\:=\:\mathrm{7}{i}\:+\:\mathrm{8}{j} \\ $$$$\left(\mathrm{a}\right)\:\mathrm{find}\:\mathrm{in}\:\mathrm{vector}\:\mathrm{notation}\:{F}_{\mathrm{1}} \:\mathrm{and}\:{F}_{\mathrm{2}} \\ $$$$\mathrm{then}\:\mathrm{find}\:\mathrm{thier}\:\mathrm{resultant}. \\ $$
Question Number 98213 Answers: 2 Comments: 1
Question Number 98205 Answers: 1 Comments: 0
$$\boldsymbol{\mathrm{Find}}\:\boldsymbol{\mathrm{the}}\:\:\boldsymbol{\mathrm{nth}}\:\:\boldsymbol{\mathrm{term}}\:\:\boldsymbol{\mathrm{of}}\:\boldsymbol{\mathrm{the}}\:\boldsymbol{\mathrm{sequence}}\:\:\left\{\boldsymbol{\mathrm{a}}_{\boldsymbol{\mathrm{n}}} \right\}\:\:\boldsymbol{\mathrm{such}}\:\boldsymbol{\mathrm{that}} \\ $$$$\boldsymbol{\mathrm{a}}_{\mathrm{1}} \:\:=\:\:\mathrm{1},\:\:\:\:\boldsymbol{\mathrm{a}}_{\boldsymbol{\mathrm{n}}\:\:+\:\:\mathrm{1}} \:\:=\:\:\frac{\mathrm{1}}{\mathrm{2}}\boldsymbol{\mathrm{a}}_{\boldsymbol{\mathrm{n}}} \:\:+\:\:\frac{\boldsymbol{\mathrm{n}}^{\mathrm{2}} \:−\:\mathrm{2}\boldsymbol{\mathrm{n}}\:\:−\:\:\mathrm{1}}{\boldsymbol{\mathrm{n}}^{\mathrm{2}} \left(\boldsymbol{\mathrm{n}}\:\:+\:\:\mathrm{1}\right)^{\mathrm{2}} }\:\:\:\:\left(\boldsymbol{\mathrm{n}}\:\:=\:\:\mathrm{1},\:\:\mathrm{2},\:\:\mathrm{3},\:\:...\right) \\ $$
Question Number 98201 Answers: 1 Comments: 0
Question Number 98286 Answers: 0 Comments: 0
$${justify}: \\ $$$$\underset{{x}\rightarrow+\infty} {{lim}}\:\underset{{k}\geqslant\mathrm{1}} {\sum}\frac{\left({k}−\mathrm{1}\right)!\:{sin}\left({x}−\frac{\pi}{\mathrm{2}}{k}\right)}{{x}^{{k}} }=\frac{\pi}{\mathrm{2}} \\ $$
Question Number 98192 Answers: 2 Comments: 1
Question Number 98191 Answers: 2 Comments: 0
$$\boldsymbol{\mathrm{for}}\:\boldsymbol{\mathrm{what}}\:\boldsymbol{\mathrm{value}}\:\boldsymbol{\mathrm{of}}\::\left\{\boldsymbol{\mathrm{x}}\mid\:\mathrm{0}<\boldsymbol{\mathrm{x}}<\mathrm{2}\boldsymbol{\pi}\right\}:\: \\ $$$$\:\:\:\:\:\:\:\:\:\mathrm{4}\boldsymbol{\mathrm{cosec}}^{\mathrm{2}} \boldsymbol{\mathrm{x}}−\mathrm{9}=\boldsymbol{\mathrm{cotx}} \\ $$
Question Number 98190 Answers: 1 Comments: 0
Question Number 98189 Answers: 1 Comments: 0
$$\mathrm{let}\:\xi\left(\mathrm{x}\right)\:=\sum_{\mathrm{n}=\mathrm{1}} ^{\infty} \:\frac{\mathrm{1}}{\mathrm{n}^{\mathrm{x}} } \\ $$$$\mathrm{calculate}\:\mathrm{lim}_{\mathrm{x}\rightarrow\mathrm{1}^{+} } \:\:\:\left(\mathrm{x}−\mathrm{1}\right)\xi\left(\mathrm{x}\right) \\ $$
Question Number 98188 Answers: 1 Comments: 0
$$\mathrm{let}\:\mathrm{f}\left(\mathrm{x}\right)\:=\frac{\mathrm{arctan}\left(\mathrm{2x}\right)}{\mathrm{x}+\mathrm{3}} \\ $$$$\left.\mathrm{1}\right)\:\mathrm{calculate}\:\mathrm{f}^{\left(\mathrm{n}\right)} \left(\mathrm{x}\right)\:\mathrm{snd}\:\mathrm{f}^{\left(\mathrm{n}\right)} \left(\mathrm{0}\right) \\ $$$$\left.\mathrm{2}\right)\:\mathrm{developp}\:\mathrm{f}\:\mathrm{at}\:\mathrm{integr}\:\mathrm{serie} \\ $$
Question Number 98187 Answers: 3 Comments: 0
$$\mathrm{find}\:\mathrm{arctan}\left(\mathrm{x}\right)+\mathrm{arctany}\:\:\mathrm{at}\:\mathrm{form}\:\mathrm{of}\:\mathrm{arctan} \\ $$
Question Number 98186 Answers: 0 Comments: 0
$$\mathrm{solve}\:\mathrm{xy}^{\left(\mathrm{3}\right)} \:+\mathrm{x}^{\mathrm{2}} \mathrm{y}^{\left(\mathrm{2}\right)} \:+\mathrm{x}^{\mathrm{3}} \mathrm{y}^{\left(\mathrm{1}\right)} \:+\mathrm{x}^{\mathrm{4}} \mathrm{y}\:=\mathrm{e}^{−\mathrm{2x}} \\ $$
Question Number 98185 Answers: 1 Comments: 0
$$\mathrm{developp}\:\mathrm{at}\:\mathrm{fourier}\:\mathrm{serie} \\ $$$$\mathrm{g}\left(\mathrm{x}\right)\:=\frac{\mathrm{2}}{\mathrm{3}+\mathrm{sin}^{\mathrm{2}} \mathrm{x}} \\ $$
Question Number 98184 Answers: 1 Comments: 0
$$\mathrm{let}\:\mathrm{f}\left(\mathrm{x}\right)\:=\int_{\mathrm{0}} ^{\infty} \:\:\frac{\mathrm{cos}\left(\mathrm{xt}\right)}{\left(\mathrm{t}^{\mathrm{2}} \:+\mathrm{4}\right)^{\mathrm{2}} }\mathrm{dt}\:\:\mathrm{find}\:\int_{\mathrm{0}} ^{\mathrm{1}} \:\mathrm{f}\left(\mathrm{x}\right)\mathrm{dx} \\ $$
Question Number 98183 Answers: 0 Comments: 0
$$\mathrm{solve}\:\mathrm{xy}^{''} \:−\frac{\mathrm{3}}{\mathrm{x}+\mathrm{1}}\mathrm{y}^{'} \:=\mathrm{xsin}\left(\mathrm{x}\right) \\ $$
Question Number 98182 Answers: 2 Comments: 0
$$\mathrm{find}\:\int\:\mathrm{x}^{\mathrm{2}} \sqrt{\frac{\mathrm{2}−\mathrm{x}}{\mathrm{2}+\mathrm{x}}}\mathrm{dx} \\ $$
Question Number 98181 Answers: 0 Comments: 0
$$\mathrm{calculate}\:\int_{\mathrm{0}} ^{\pi} \mathrm{ln}\left(\mathrm{x}^{\mathrm{2}} −\mathrm{2xcos}\theta\:+\mathrm{1}\right)\mathrm{d}\theta \\ $$
Question Number 98180 Answers: 0 Comments: 0
$$\mathrm{solve}\:\:\mathrm{xy}^{''} \:+\left(\mathrm{x}^{\mathrm{3}} +\mathrm{1}\right)\mathrm{y}\:=\mathrm{3e}^{\mathrm{2x}} \\ $$
Question Number 98179 Answers: 2 Comments: 0
$$\mathrm{calculate}\:\int_{\mathrm{1}} ^{+\infty} \:\:\:\frac{\mathrm{dx}}{\mathrm{x}^{\mathrm{2}} \left(\mathrm{1}−\mathrm{x}^{\mathrm{2}} \right)^{\mathrm{3}} } \\ $$
Question Number 98178 Answers: 0 Comments: 0
$$\mathrm{calculate}\:\:\sum_{\mathrm{p}=\mathrm{0}} ^{\mathrm{n}} \:\left(\mathrm{z}+\mathrm{1}\right)^{\mathrm{p}} \\ $$$$\mathrm{with}\:\mathrm{z}\:\mathrm{root}\:\mathrm{of}\:\mathrm{x}^{\mathrm{2}} −\mathrm{x}+\mathrm{1}=\mathrm{0} \\ $$
Question Number 98290 Answers: 0 Comments: 1
Question Number 98166 Answers: 0 Comments: 0
$$\underset{{x}\rightarrow\infty} {{lim}}\left[{cos}\left(\mathrm{2}\pi\left(\frac{{x}}{{x}+\mathrm{1}}\right)^{{a}} \right)\right]^{{x}^{\mathrm{2}} } \\ $$$${a}\in\mathbb{R} \\ $$
Question Number 98151 Answers: 2 Comments: 0
$$\int{e}^{{x}^{\mathrm{5}} +\mathrm{8}{x}^{\mathrm{2}} } {dx} \\ $$$$=\frac{\sqrt{\pi}}{\mathrm{4}\sqrt{\mathrm{2}}}{e}^{{x}^{\mathrm{5}} } {erfi}\left(\mathrm{2}\sqrt{\mathrm{2}}{x}\right)−\frac{\mathrm{5}\sqrt{\pi}}{\mathrm{4}\left(\mathrm{128}\right)\sqrt{\mathrm{2}}}\left({super}−{erf}_{\left({hyper}\right)} \left(\mathrm{2}\sqrt{\mathrm{2}}{x}\right)\right)+{c} \\ $$$$ \\ $$$${where}\left[{super}−{erf}_{\left({hyper}\right)} \left({t}\right)\right]\:{is}\:{super}−{function} \\ $$$${in}\:{D}_{\mathrm{2}} \:{and}\:\left[{D}_{{n}} \right] \\ $$
Question Number 98149 Answers: 1 Comments: 1
Question Number 98148 Answers: 0 Comments: 0
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