Question Number 223400 by behi834171 last updated on 23/Jul/25 $$\boldsymbol{{f}}\left(\mathrm{1}\right)=\mathrm{2025} \\ $$$$\underset{\mathrm{1}} {\overset{\boldsymbol{{n}}} {\boldsymbol{\sum}}{f}}\left(\boldsymbol{{k}}\right)=\boldsymbol{{n}}^{\mathrm{2}} .\boldsymbol{{f}}\left(\boldsymbol{{n}}\right) \\ $$$$\boldsymbol{{f}}\left(\mathrm{2025}\right)=? \\ $$ Answered by mr W last updated…
Question Number 223386 by behi834171 last updated on 23/Jul/25 Answered by mr W last updated on 23/Jul/25 $${still}\:{using}\:{my}\:{method}\:{generally}\:{for} \\ $$$$\boldsymbol{{x}}^{\boldsymbol{{n}}} +\boldsymbol{{kx}}+\boldsymbol{{p}}=\mathrm{0} \\ $$$${x}^{{n}} −{a}=−{k}\left({x}+\frac{{p}+{a}}{{k}}\right)=−{ky} \\…
Question Number 223383 by mr W last updated on 23/Jul/25 Commented by Frix last updated on 23/Jul/25 $$\mathrm{I}\:\mathrm{guess}\:\mathrm{it}'\mathrm{s}\:\mathrm{66} \\ $$ Commented by Frix last updated…
Question Number 223374 by behi834171 last updated on 22/Jul/25 Answered by Ghisom last updated on 22/Jul/25 $${r}^{\mathrm{4}} +\mathrm{4}{r}+\mathrm{1}=\mathrm{0} \\ $$$${r}^{\mathrm{4}} =−\mathrm{4}{r}−\mathrm{1} \\ $$$${r}^{\mathrm{4}} −\mathrm{1}=−\mathrm{4}{r}−\mathrm{2} \\…
Question Number 223368 by Nicholas666 last updated on 22/Jul/25 $$ \\ $$$$\:\:\:\:\:\:\:\:\int_{\mathrm{0}} ^{\mathrm{1}} \:\boldsymbol{\mathrm{ln}}\left(\frac{\mathrm{2}\:\boldsymbol{\mathrm{cos}}\left({x}^{\mathrm{2}} \right)\:+\:\boldsymbol{\mathrm{ln}}^{\mathrm{2}} \left({x}/\mathrm{2}\right)}{\mathrm{1}\:+\:\boldsymbol{\mathrm{cos}}\:\left({x}/\mathrm{2}\right)}\right)\:\boldsymbol{\mathrm{d}}{x} \\ $$$$ \\ $$ Answered by MathematicalUser2357 last updated…
Question Number 223354 by ajfour last updated on 22/Jul/25 Commented by ajfour last updated on 22/Jul/25 https://g.co/gemini/share/a41e606602a3 Commented by ajfour last updated on 22/Jul/25 Commented…
Question Number 223367 by Nicholas666 last updated on 22/Jul/25 $$ \\ $$$$\:\:\:\:\:\:\:\:\:\int_{\mathrm{0}} ^{\mathrm{1}} \:\:\mathrm{ln}\left(\mathrm{2}\:\mathrm{cos}\left({x}^{\mathrm{2}} \right)\:+\:\mathrm{ln}^{\mathrm{2}} \:\left(\frac{{x}}{\mathrm{2}}\right)\:\mathrm{d}{x}\right. \\ $$$$ \\ $$ Commented by MathematicalUser2357 last updated…
Question Number 223340 by fantastic last updated on 21/Jul/25 Commented by fantastic last updated on 21/Jul/25 $${This}\:{will}\:{take}\:{a}\:{lot}\:{of}\:{time} \\ $$ Answered by A5T last updated on…
Question Number 223304 by Nadirhashim last updated on 21/Jul/25 $$\:\:\:\underset{{x}\rightarrow\mathrm{0}} {\mathrm{lim}}\frac{\underset{\mathrm{0}} {\overset{{x}^{\mathrm{2}} } {\int}}\boldsymbol{{sin}}\left(\sqrt{\boldsymbol{{t}}}\right)\boldsymbol{{dt}}\:}{\boldsymbol{{x}}^{\mathrm{3}} }\:=…? \\ $$ Answered by Raphael254 last updated on 21/Jul/25 $$…
Question Number 223348 by cryptograph last updated on 21/Jul/25 $${Determine}\:{gcd}\left(\mathrm{13}{a}+\mathrm{19}{b},{ab}\right)\:{given}\:{that}\:{gcd}\left({a},\mathrm{19}\right)={gcd}\left({b},\mathrm{13}\right)=\mathrm{1} \\ $$ Commented by A5T last updated on 23/Jul/25 $$\mathrm{This}\:\mathrm{isn}'\mathrm{t}\:\mathrm{unique}.\:\mathrm{a}=\mathrm{1}\:\mathrm{and}\:\mathrm{b}=\mathrm{1}\Rightarrow\:\mathrm{gcd}\left(\mathrm{32},\mathrm{1}\right)=\mathrm{1} \\ $$$$\mathrm{a}=\mathrm{2}\:\mathrm{and}\:\mathrm{b}=\mathrm{2}\:\Rightarrow\:\mathrm{gcd}\left(\mathrm{64},\mathrm{4}\right)=\mathrm{4} \\ $$ Answered…