Question Number 221778 by MrGaster last updated on 10/Jun/25 $$\mathrm{For}\:\forall{n}\in\boldsymbol{{N}}^{\ast} ,{n}\geq\mathrm{3}\:\mathrm{Prove}: \\ $$$$\int_{\mathrm{0}} ^{\mathrm{1}} \left(\underset{\mathrm{2}<{p}\leq\mathrm{2}{n}} {\sum}{e}^{\mathrm{2}\pi{ip}+\alpha} \right)^{\mathrm{2}} {e}^{−\mathrm{4}\pi{in}\alpha} {d}\alpha>\mathrm{0},{p}\:\mathrm{is}\:\mathrm{a}\:\mathrm{prime}\:\mathrm{number} \\ $$ Terms of Service Privacy…
Question Number 221769 by MrGaster last updated on 10/Jun/25 $$\int_{\mathrm{0}} ^{\pi} \frac{{a}}{{a}−\mathrm{cos}^{\mathrm{2}{n}} {x}}{dx}=?,{a}>\mathrm{1} \\ $$$$\int_{\mathrm{0}} ^{\pi} \frac{\mathrm{2}}{\mathrm{2}−\mathrm{cos}^{\mathrm{4}} {x}}{dx}=? \\ $$$$\underset{{m}\rightarrow\infty} {\mathrm{lim}}\int_{\mathrm{0}} ^{\pi} \frac{\mathrm{cos}^{\mathrm{2}{n}} \left(\mathrm{2}{mx}\right)}{{a}−\mathrm{cos}^{\mathrm{2}{n}} {x}}{dx}=?,{a}>\mathrm{1},{n}\in\mathbb{N}^{+}…
Question Number 221770 by MrGaster last updated on 10/Jun/25 $$\mathrm{Prove}:\int_{\mathrm{0}} ^{\mathrm{1}} \frac{\mathrm{arcsin}{x}}{\mathrm{1}+{x}^{\mathrm{4}} }{dx}=\frac{\sqrt{\mathrm{2}}\pi^{\mathrm{2}} }{\mathrm{16}}−\frac{\sqrt{\mathrm{2}}\pi}{\mathrm{8}}\mathrm{ln}\left(\sqrt{\mathrm{2}}−\mathrm{1}\right)+\mathrm{2}\underset{{n}=\mathrm{0}} {\overset{\infty} {\sum}}\frac{\left(−\right)^{{n}} }{\left(\mathrm{2}{n}+\mathrm{1}\right)}\mid{z}_{\mathrm{0}} \mid^{\mathrm{2}{n}+\mathrm{1}} \mathrm{sin}\left(\frac{\pi}{\mathrm{4}}−\left(\mathrm{2}{n}+\mathrm{1}\right)\beta\right) \\ $$ Terms of Service Privacy…
Question Number 221661 by mr W last updated on 09/Jun/25 Commented by mr W last updated on 09/Jun/25 $${an}\:{unsolved}\:{old}\:{question} \\ $$ Answered by MrGaster last…
Question Number 221663 by MrGaster last updated on 09/Jun/25 $$\mathrm{Prove}:\int_{\mathrm{0}} ^{\mathrm{1}} \underset{{k}=\mathrm{1}} {\overset{\infty} {\prod}}\left(\mathrm{1}−{x}^{{k}} \right){dx}=\frac{\mathrm{4}\pi\sqrt{\mathrm{3}}}{\:\sqrt{\mathrm{23}}}\centerdot\frac{\mathrm{sinh}\frac{\sqrt{\mathrm{23}}\pi}{\mathrm{6}}}{\mathrm{2}\:\mathrm{cosh}\frac{\sqrt{\mathrm{23}}\pi}{\mathrm{3}}−\mathrm{1}} \\ $$ Commented by MrGaster last updated on 09/Jun/25 It is difficult for me to give an analytical solution to that integral.…
Question Number 221721 by ahmed2025 last updated on 09/Jun/25 Answered by wewji12 last updated on 09/Jun/25 $$\frac{\mathrm{d}{y}}{\mathrm{dln}\left({t}\right)}=\frac{\frac{\mathrm{d}{y}}{\mathrm{d}{t}}}{\frac{\mathrm{d}\left\{\mathrm{ln}\left({t}\right)\right\}}{\mathrm{d}{t}}}\:\left(\mathrm{Warning}\:\:\frac{\mathrm{d}{y}}{\mathrm{d}{x}}\:\mathrm{is}\:\mathrm{NOT}\:\mathrm{Fraction}!!!\right) \\ $$$$\frac{\frac{\mathrm{d}\:\:}{\mathrm{d}{t}}\left\{{t}^{\mathrm{2}} \mathrm{ln}\left({t}\right)\right\}}{\frac{\mathrm{d}\:\:}{\mathrm{d}{t}}\left\{\mathrm{ln}\left({t}\right)\right\}}=\frac{\mathrm{2}{t}\mathrm{ln}\left({t}\right)+{t}}{\frac{\mathrm{1}}{{t}}}={t}^{\mathrm{2}} +\mathrm{2}{t}^{\mathrm{2}} \mathrm{ln}\left({t}\right)={t}^{\mathrm{2}} \left(\mathrm{1}+\mathrm{2ln}\left({t}\right)\right) \\ $$$$\frac{\mathrm{d}{y}\left({t}\right)}{\mathrm{d}\left\{\mathrm{ln}\left({t}\right)\right\}}={t}^{\mathrm{2}}…
Question Number 221754 by fantastic last updated on 09/Jun/25 $$\left.\sqrt[{\sqrt[{\sqrt[{\:\mathrm{3}^{\mathrm{4}^{\mathrm{0}^{\mathrm{4}^{\mathrm{3}} } } } }]{\mathrm{27}}}]{\mathrm{64}}}]{\mathrm{81}}\right)^{\sqrt{\mathrm{4}}} \\ $$ Answered by wewji12 last updated on 09/Jun/25 $$…..\mathrm{what}\:\mathrm{a}\:\mathrm{horrible}\:\mathrm{notation} \\…
Question Number 221686 by Tawa11 last updated on 09/Jun/25 Commented by AlagaIbile last updated on 09/Jun/25 $$\:{Let}\:\mathrm{tan}\:\boldsymbol{{x}}\:=\:\boldsymbol{{y}},\:\mathrm{cos}\:\boldsymbol{{x}}\:=\:\frac{\mathrm{1}}{\:\sqrt{\boldsymbol{{y}}^{\mathrm{2}} \:+\:\mathrm{1}}} \\ $$$$\Rightarrow\:\mathrm{cos}^{\mathrm{2}\:} \left[\mathrm{cos}^{-\mathrm{1}} \:\frac{\mathrm{1}}{\:\sqrt{\boldsymbol{{y}}^{\mathrm{2}} \:+\:\mathrm{1}}}\right] \\ $$$$\Rightarrow\:\left[\mathrm{cos}\:\left(\mathrm{cos}^{-\mathrm{1}}…
Question Number 221680 by ajfour last updated on 09/Jun/25 $${I}\:{suspect} \\ $$$$\pi={i}\underset{\boldsymbol{{i}}} {\overset{−\mathrm{1}} {\int}}\frac{\left({z}−\mathrm{1}\right){dz}}{\:{z}\sqrt{{z}^{\mathrm{2}} +\mathrm{1}}} \\ $$$${someone}\:{please}\:{help}\:{confirm}\:{or}\:{reject}! \\ $$ Commented by fantastic last updated on…
Question Number 221647 by Jubr last updated on 09/Jun/25 $${solve}\:{for}\:{x}. \\ $$$${x}^{\mathrm{1}} \:+\:{x}^{\mathrm{2}} \:+\:{x}^{\mathrm{3}} \:\:=\:\:\mathrm{4096} \\ $$ Commented by Frix last updated on 09/Jun/25 $${x}^{\mathrm{3}}…