Question Number 196544 by mr W last updated on 27/Aug/23 Answered by witcher3 last updated on 27/Aug/23 $$\mathrm{f}\left(\mathrm{k}+\mathrm{1}\right)=\mathrm{f}\left(\mathrm{1}\right)+\mathrm{f}\left(\mathrm{k}\right)+\mathrm{k} \\ $$$$\Rightarrow\underset{\mathrm{k}=\mathrm{1}} {\overset{\mathrm{n}−\mathrm{1}} {\sum}}\left(\mathrm{f}\left(\mathrm{k}+\mathrm{1}\right)−\mathrm{f}\left(\mathrm{k}\right)\right)=\underset{\mathrm{k}=\mathrm{1}} {\overset{\mathrm{n}−\mathrm{1}} {\sum}}\mathrm{f}\left(\mathrm{1}\right)+\mathrm{k}=\left(\mathrm{n}−\mathrm{1}\right)\mathrm{f}\left(\mathrm{1}\right)+\frac{\mathrm{n}}{\mathrm{2}}\left(\mathrm{n}−\mathrm{1}\right) \\…
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Question Number 196496 by RoseAli last updated on 26/Aug/23 Answered by witcher3 last updated on 26/Aug/23 $$\int\frac{\mathrm{1}}{\mathrm{tg}^{\mathrm{2}} \left(\mathrm{x}\right).\mathrm{cos}^{\mathrm{6}} \left(\mathrm{x}\right)} \\ $$$$=\int\frac{\mathrm{1}+\mathrm{tg}^{\mathrm{2}} \left(\mathrm{x}\right)}{\mathrm{tg}^{\mathrm{2}} \left(\mathrm{x}\right)}.\left(\mathrm{1}+\mathrm{tg}^{\mathrm{2}} \left(\mathrm{x}\right)\right)^{\mathrm{2}} \mathrm{dx}…
Question Number 196498 by Pedro2002 last updated on 26/Aug/23 Answered by Frix last updated on 26/Aug/23 $$\int\mathrm{e}^{−{x}^{\mathrm{2}} } {dx}=\frac{\sqrt{\pi}}{\mathrm{2}}\int\frac{\mathrm{2}}{\:\sqrt{\pi}}\mathrm{e}^{−{x}^{\mathrm{2}} } {dx}=\frac{\sqrt{\pi}}{\mathrm{2}}\mathrm{erf}\:{x}\:+{C} \\ $$ Terms of…
Question Number 196525 by Mastermind last updated on 26/Aug/23 Answered by mr W last updated on 27/Aug/23 $$\sqrt{{m}^{\mathrm{2}} +\mathrm{1}}>\sqrt{{m}^{\mathrm{2}} }={m} \\ $$$$\sqrt{{m}^{\mathrm{2}} +\mathrm{1}}<\sqrt{{m}^{\mathrm{2}} +\mathrm{2}{m}+\mathrm{1}}=\sqrt{\left({m}+\mathrm{1}\right)^{\mathrm{2}} }={m}+\mathrm{1}…
Question Number 196493 by mr W last updated on 26/Aug/23 Commented by mr W last updated on 27/Aug/23 $${solution}\:{to}\:{question}\:#\mathrm{196172} \\ $$ Answered by mr W…
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Question Number 196523 by ERLY last updated on 27/Aug/23 $${resoudre}\:{dans}\:{c}\:{l}\:{equation}\:{sinx}=\mathrm{2}\:\:\:\:\:\bigstar{erly}\:{rolvinst}\bigstar\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:<{erly}\:{rolvinst}> \\ $$ Answered by Frix last updated on 26/Aug/23 $$\mathrm{sin}\:{x}\:=\frac{\mathrm{e}^{\mathrm{i}{x}} −\mathrm{e}^{−\mathrm{i}{x}} }{\mathrm{2i}}=\mathrm{2} \\ $$$$\mathrm{e}^{\mathrm{i}{x}} −\frac{\mathrm{1}}{\mathrm{e}^{\mathrm{i}{x}}…
Question Number 196491 by universe last updated on 26/Aug/23 Commented by hardmath last updated on 26/Aug/23 $$=\:\mathrm{2}^{\frac{\mathrm{2}}{\boldsymbol{\pi}}} \: \\ $$ Commented by universe last updated…