Question Number 219267 by zetamaths last updated on 21/Apr/25 $${Une}\:{fonction}\:{P}\:{est}\:{dite}\:{quasi}\:{polynomiale}\:{s}'{il}\:{existe}\:\left({pour}\:{k}\in\mathbb{N}\:\right)\:{k}+\mathrm{1}\:{fonction}\:{periodique}\left({c}_{{i}} \right)_{{i}\in\left[\mid\mathrm{0};{k}\mid\right]} {de}\:\mathbb{Z}\:{dans}\:\mathbb{R} \\ $$$$\:{telles}\:{que}\:{P}\left({n}\right)=\underset{{k}=\mathrm{1}} {\overset{{n}} {\sum}}{c}_{{i}} \left({n}\right){n}^{{i}} \\ $$$$\left(\mathrm{1}\right)\:{Montrez}\:{que}\:{l}'{ensemble}\:{des}\:{fonction}\:{quasi}\:{polynomiale}\:{forme}\:{un}\:\mathbb{R}−{ev}\left({real}\:{space}\:{vector}\right). \\ $$$$\left(\mathrm{2}\right){Montrez}\:{que}\:{si}\:{P},{Q}:\mathbb{Z}\rightarrow\mathbb{R}\:{sont}\:{desfonction}\:{quasi}\:{polynomiale}\:{tel}\:{que}\:{P}\left({n}\right)={Q}\left({n}\right)\:\forall{n}\in\mathbb{N}\:{alors}\:{P}={Q} \\ $$ Answered by…
Question Number 219135 by Rojarani last updated on 20/Apr/25 Answered by Frix last updated on 20/Apr/25 $${a}^{\frac{\mathrm{1}}{\mathrm{3}}} +{b}^{\frac{\mathrm{1}}{\mathrm{3}}} ={c}^{\frac{\mathrm{1}}{\mathrm{3}}} \\ $$$${a}+{b}+\mathrm{3}{a}^{\frac{\mathrm{1}}{\mathrm{3}}} {b}^{\frac{\mathrm{1}}{\mathrm{3}}} \underset{={c}^{\frac{\mathrm{1}}{\mathrm{3}}} } {\underbrace{\left({a}^{\frac{\mathrm{1}}{\mathrm{3}}}…
Question Number 219193 by fantastic last updated on 20/Apr/25 $$\int\sqrt{\frac{{x}+\mathrm{1}}{{x}+\mathrm{2}}}\:.\frac{\mathrm{1}}{{x}+\mathrm{3}}\:{dx}=? \\ $$ Answered by aleks041103 last updated on 20/Apr/25 $$\frac{{x}+\mathrm{1}}{{x}+\mathrm{2}}={u}=\mathrm{1}−\frac{\mathrm{1}}{{x}+\mathrm{2}} \\ $$$$\Rightarrow{x}=\frac{\mathrm{1}}{\mathrm{1}−{u}}−\mathrm{2}=\frac{\mathrm{2}{u}−\mathrm{1}}{\mathrm{1}−{u}} \\ $$$$\Rightarrow{dx}=\frac{\mathrm{2}\left(\mathrm{1}−{u}\right)−\left(−\mathrm{1}\right)\left(\mathrm{2}{u}−\mathrm{1}\right)}{\left(\mathrm{1}−{u}\right)^{\mathrm{2}} }{du}=…
Question Number 219190 by fantastic last updated on 20/Apr/25 Commented by AntonCWX8 last updated on 21/Apr/25 $${Youtube}\:{has}\:{a}\:{lot}\:{of}\:{videos}\:{explaining}\:{this}. \\ $$$${Search}\:{it}\:{yourself}. \\ $$ Answered by Marzuk last…
Question Number 219185 by fantastic last updated on 20/Apr/25 Answered by MrGaster last updated on 20/Apr/25 $$\bigtriangledown\centerdot\left(\overset{\rightarrow} {{F}}×\overset{\rightarrow} {{G}}\right)=\partial_{{i}} \left(\epsilon_{{ijk}} {F}_{{j}} {G}_{{k}} \right) \\ $$$$=\epsilon_{{ijk}}…
Question Number 219181 by hardmath last updated on 20/Apr/25 Answered by devdutt last updated on 20/Apr/25 $$\underset{{k}=\mathrm{1}} {\overset{{n}} {\sum}}\frac{{k}^{\mathrm{2}} }{\mathrm{2}{k}^{\mathrm{2}} −\mathrm{2}{nk}+{n}^{\mathrm{2}} }\:=\:\underset{{k}=\mathrm{0}} {\overset{{n}} {\sum}}\frac{{k}^{\mathrm{2}} }{{k}^{\mathrm{2}}…
Question Number 219182 by skcusb last updated on 20/Apr/25 Commented by MathematicalUser2357 last updated on 22/Apr/25 $$\int{y}^{\mathrm{2}} \left({y}+\mathrm{2}\right)^{\frac{\mathrm{1}}{\mathrm{4}}} {dy} \\ $$ Answered by MrGaster last…
Question Number 219236 by Nicholas666 last updated on 20/Apr/25 $$ \\ $$$$\:\:\:\:{f}\left({t}\right)\:=\:\frac{\mathrm{1}}{\mathrm{2}\pi{i}}\:\int_{\:{c}−{i}\infty} ^{\:{c}+{i}\infty} \:\frac{{e}^{{st}} }{{s}^{{k}} \:}\:\:{ds}\:\:\:,\:\:{k}\:\in\mathbb{C} \\ $$$$\: \\ $$ Commented by Nicholas666 last updated…
Question Number 219173 by Spillover last updated on 20/Apr/25 Answered by Spillover last updated on 20/Apr/25 R²= 9² + (16+x)² R² = 21² + x²…
Question Number 219232 by Spillover last updated on 20/Apr/25 Answered by A5T last updated on 21/Apr/25 Commented by A5T last updated on 21/Apr/25 $$\mathrm{AD}=\mathrm{bsin}\theta\:;\:\mathrm{AE}=\mathrm{bcos}\theta\:;\:\mathrm{AC}=\mathrm{asin}\theta\:;\:\mathrm{AB}=\mathrm{acos}\theta \\…