Question Number 222249 by MrGaster last updated on 21/Jun/25 $$\mathrm{Prove}:\forall{n}\in\mathbb{Z}^{+} ,\mathrm{1}^{\mathrm{3}} +\mathrm{2}^{\mathrm{3}} +\ldots+{n}^{\mathrm{3}} =\left(\mathrm{1}+\mathrm{2}+\ldots+{n}\right)^{\mathrm{2}} \\ $$ Answered by MrGaster last updated on 21/Jun/25 Answered by…
Question Number 222276 by Nicholas666 last updated on 27/Jun/25 $$ \\ $$$$\:\mathrm{Prove};\:\underset{{k}\:=\:\mathrm{1}} {\overset{\infty} {\sum}}\:\frac{{k}}{\mathrm{sinh}\left(\pi{k}\right)}\:=\:\frac{\Gamma\left(\frac{\mathrm{1}}{\mathrm{4}}\right)^{\mathrm{4}} −\:\mathrm{8}\pi^{\mathrm{2}} }{\mathrm{32}\pi^{\mathrm{3}} }\:\:\:\:\:\:\:\:\:\:\:\:\:\: \\ $$$$ \\ $$ Terms of Service Privacy…
Question Number 222279 by fantastic last updated on 21/Jun/25 $$\left({a}^{\mathrm{2}} −{b}^{\mathrm{2}} \right)\mathrm{sin}\:\theta+\mathrm{2}{ab}\mathrm{cos}\:\theta={a}^{\mathrm{2}} +{b}^{\mathrm{2}} \\ $$$$\mathrm{tan}\:\theta=?? \\ $$ Answered by mr W last updated on 21/Jun/25…
Question Number 222275 by Nicholas666 last updated on 21/Jun/25 $$ \\ $$$$\:\:\mathrm{Prove}\:;\:\int_{−\pi} ^{\:\pi} \:\frac{{z}\:\mathrm{sin}\left({z}\right)\:}{\left(\mathrm{1}\:+\:{z}\:+\:\sqrt{\mathrm{1}\:+\:{z}^{\mathrm{2}} }\right)\sqrt{\mathrm{3}\:+\:\mathrm{sin}^{\mathrm{2}} \left({z}\right)}}\:{dz}\:=\:\zeta\left(\mathrm{2}\right)\:\:\:\:\:\:\:\: \\ $$$$ \\ $$ Answered by MrGaster last updated…
Question Number 222197 by BHOOPENDRA last updated on 20/Jun/25 $${question}\:\mathrm{211277} \\ $$ Answered by BHOOPENDRA last updated on 20/Jun/25 Commented by BHOOPENDRA last updated on…
Question Number 222224 by MrGaster last updated on 20/Jun/25 $$\mathrm{Prove}:\int_{\mathrm{0}} ^{\mathrm{1}} \frac{\mathrm{ln}\left(\mathrm{1}−{x}^{\mathrm{2}} \right)}{{x}}\mathrm{cos}\left(\mathrm{ln}\:{x}\right){dx}=\mathrm{1}−\frac{\pi}{\mathrm{2}}\mathrm{cosh}\frac{\pi}{\mathrm{2}} \\ $$ Commented by Nicholas666 last updated on 21/Jun/25 $$\:\mathrm{i}\:\mathrm{think}\:\mathrm{the}\:\mathrm{statment}\:\mathrm{is}\:\mathrm{wrong},\: \\ $$$$\mathrm{no}\:\mathrm{1}−\frac{\pi}{\mathrm{2}}\:\mathrm{cosh}\:\frac{\pi}{\mathrm{2}}\:\mathrm{but}\:\mathrm{1}\:−\frac{\pi}{\mathrm{2}}\mathrm{chot}\:\left(\frac{\pi}{\mathrm{2}}\right)…
Question Number 222192 by mr W last updated on 22/Jun/25 $${there}\:{are}\:\mathrm{32}\:{students}\:{in}\:{a}\:{class}.\:{for} \\ $$$${each}\:{competition}\:{in}\:{a}\:{sport}\:{event}\: \\ $$$${in}\:{the}\:{school}\:{each}\:{class}\:{can}\:{send} \\ $$$${a}\:{team}\:{with}\:{three}\:{students}.\:{if}\:{no} \\ $$$${two}\:{students}\:{may}\:{be}\:{in}\:{the}\:{same} \\ $$$${team}\:{for}\:{more}\:{than}\:{one}\:{time},\:{in} \\ $$$${how}\:{many}\:{different}\:{competitions}\: \\ $$$${can}\:{this}\:{class}\:{participate}?…
Question Number 222225 by mathBagja last updated on 21/Jun/25 $${a}+{b}+{c}=−{D} \\ $$$${ab}+{bc}+{ac}={E} \\ $$$${abc}=−\mathrm{1} \\ $$$$\left({a}^{\mathrm{2}} −{bc}\right)\left({b}^{\mathrm{2}} −{ac}\right)\left({c}^{\mathrm{2}} −{ab}\right)=\mathrm{40} \\ $$$${D}−{E}=\mathrm{4} \\ $$$$\left({D}+{E}\right)^{\mathrm{2}} =… \\…
Question Number 222193 by MrGaster last updated on 20/Jun/25 $$\mathrm{Prove}: \\ $$$$\frac{\mathrm{1}}{\mathrm{sin}\left(\frac{\mathrm{3}\pi}{\mathrm{8}}\right)}=\mathrm{2}\left(\mathrm{cos}\left(\frac{\pi}{\mathrm{8}}\right)−\mathrm{sin}\left(\frac{\pi}{\mathrm{8}}\right)\right) \\ $$$$\frac{\mathrm{2}}{\mathrm{sin}\left(\frac{\mathrm{8}\pi}{\mathrm{9}}\right)}−\frac{\mathrm{1}}{\mathrm{sin}\left(\frac{\mathrm{5}\pi}{\mathrm{9}}\right)}=\mathrm{2}\sqrt{\mathrm{3}}+\mathrm{4sin}\left(\frac{\pi}{\mathrm{9}}\right) \\ $$$$\frac{\mathrm{2}}{\mathrm{sin}\left(\frac{\mathrm{6}\pi}{\mathrm{7}}\right)}+\frac{\mathrm{1}}{\mathrm{sin}\left(\frac{\mathrm{4}\pi}{\mathrm{7}}\right)}=\mathrm{4}\left(\mathrm{sin}\left(\frac{\pi}{\mathrm{7}}\right)+\mathrm{cos}\left(\frac{\pi}{\mathrm{14}}\right)\right) \\ $$ Answered by som(math1967) last updated on 20/Jun/25…
Question Number 222195 by MathematicalUser2357 last updated on 20/Jun/25 $${f}\left({x}\right)={x}^{\mathrm{3}} +\sqrt{{x}}+\mathrm{sin}\:{x} \\ $$$${f}=? \\ $$ Answered by MrGaster last updated on 20/Jun/25 Commented by MathematicalUser2357…