Question Number 162520 by HongKing last updated on 30/Dec/21 $$\mathrm{Find}: \\ $$$$\boldsymbol{\Omega}\:=\underset{\:\mathrm{0}} {\overset{\:\boldsymbol{\pi}} {\int}}\:\left(\frac{\mathrm{x}\:\mathrm{cos}\:\mathrm{x}}{\mathrm{1}\:+\:\mathrm{sin}\:\mathrm{x}}\right)^{\mathrm{2}} \mathrm{dx}\: \\ $$ Answered by Ar Brandon last updated on 30/Dec/21…
Question Number 162522 by HongKing last updated on 30/Dec/21 $$\mathrm{Determine}\:\mathrm{all}\:\mathrm{positive}\:\mathrm{integers} \\ $$$$\boldsymbol{\mathrm{N}}\:\mathrm{which}\:\mathrm{the}\:\mathrm{sphere} \\ $$$$\mathrm{x}^{\mathrm{2}} \:+\:\mathrm{y}^{\mathrm{2}} \:+\:\mathrm{z}^{\mathrm{2}} \:=\:\mathrm{N} \\ $$$$\mathrm{has}\:\mathrm{an}\:\mathrm{inseribed}\:\mathrm{regular}\:\mathrm{tetrahedron} \\ $$$$\mathrm{whose}\:\mathrm{vertices}\:\mathrm{have}\:\mathrm{integer}\:\mathrm{coordinates} \\ $$ Answered by…
Question Number 162506 by amin96 last updated on 29/Dec/21 Commented by MJS_new last updated on 30/Dec/21 $$\mathrm{no}\:\mathrm{solution}\:\in\mathbb{C} \\ $$ Terms of Service Privacy Policy Contact:…
Question Number 96951 by aurpeyz last updated on 05/Jun/20 $${prove}\:{that}\:\mathrm{1}−\frac{\mathrm{1}}{\mathrm{2}}+\frac{\mathrm{1}}{\mathrm{3}}−\frac{\mathrm{1}}{\mathrm{4}}+\frac{\mathrm{1}}{\mathrm{5}}+…+\frac{−\mathrm{1}^{{n}−\mathrm{1}} }{{n}}\:\:{is}\:{always}\:{positive} \\ $$$$ \\ $$ Commented by mr W last updated on 05/Jun/20 $${how}\:{many}\:{times}\:{do}\:{you}\:{want}\:{to}\:{post} \\…
Question Number 31409 by rahul 19 last updated on 08/Mar/18 $${The}\:{maximum}\:{area}\:{of}\:{the}\:{triangle} \\ $$$${whose}\:{sides}\:{a},{b}\:{and}\:{c}\:{satisfy}\: \\ $$$$\mathrm{0}\leqslant{a}\leqslant\mathrm{1}\:,\:\mathrm{1}\leqslant{b}\leqslant\mathrm{2}\:,\:\mathrm{2}\leqslant{c}\leqslant\mathrm{3}\:{is}\:: \\ $$$$\left.{A}\right)\:\mathrm{1} \\ $$$$\left.{B}\right)\:\mathrm{2} \\ $$$$\left.{C}\right)\:\mathrm{1}.\mathrm{5} \\ $$$$\left.{D}\right)\:\mathrm{0}.\mathrm{5}\:\:\:\:\:\:\:? \\ $$…
Question Number 162478 by HongKing last updated on 29/Dec/21 $$\mathrm{Calculate}:\:\:\underset{\boldsymbol{\mathrm{k}}=\mathrm{1}} {\overset{\infty} {\sum}}\:\frac{\mathrm{H}_{\boldsymbol{\mathrm{k}}} \:\mathrm{2}^{-\boldsymbol{\mathrm{k}}} }{\mathrm{k}\:+\:\mathrm{1}} \\ $$$$\mathrm{where}\:\mathrm{H}_{\boldsymbol{\mathrm{k}}} \:\mathrm{is}\:\mathrm{the}\:\boldsymbol{\mathrm{k}}-\mathrm{th}\:\mathrm{harmonic}\:\mathrm{number} \\ $$ Answered by mnjuly1970 last updated on…
Question Number 162429 by Ari last updated on 29/Dec/21 $${put}\:{the}\:{digits}\:\mathrm{0},\mathrm{1},\mathrm{2},\mathrm{3},\mathrm{4},\mathrm{5},\mathrm{6},\mathrm{7},\mathrm{8},\mathrm{9},{in}\:{place}\:{of}\:{the}\:{letters}\:{in}\:{order}\:{to}\:{perform}\:{the}\:{edditon} \\ $$ Commented by Ari last updated on 29/Dec/21 Terms of Service Privacy Policy Contact:…
Question Number 162416 by HongKing last updated on 29/Dec/21 $$\mathrm{Prove}\:\mathrm{that}: \\ $$$$\underset{\:\mathrm{0}} {\overset{\:\frac{\boldsymbol{\pi}}{\mathrm{4}}} {\int}}\:\frac{\mathrm{4}\:\mathrm{ln}\:\left(\mathrm{cot}\boldsymbol{\mathrm{x}}\right)}{\mathrm{cos}\left(\mathrm{2x}\:+\:\mathrm{2022}\boldsymbol{\pi}\right)}\:\mathrm{dx}\:=\:\mathrm{3}\boldsymbol{\zeta}\left(\mathrm{2}\right) \\ $$ Commented by smallEinstein last updated on 29/Dec/21 Answered by…
Question Number 162417 by HongKing last updated on 29/Dec/21 $$\mathrm{Prove} \\ $$$$\underset{\:\mathrm{0}} {\overset{\:\frac{\boldsymbol{\pi}}{\mathrm{2}}} {\int}}\:\mathrm{sin}^{\mathrm{8}} \left(\mathrm{x}\right)\:+\:\underset{\:\mathrm{0}} {\overset{\:\mathrm{1}} {\int}}\:\mathrm{sin}^{-\mathrm{1}} \:\left(\sqrt[{\mathrm{8}}]{\mathrm{x}}\right)\:\geqslant\:\frac{\pi}{\mathrm{2}} \\ $$ Commented by mr W last…
Prove-the-Identity-for-any-a-n-in-Real-Number-1-a-a-n-a-a-2-n-2-a-2-n-1-2-Greatest-Integer-Function-
Question Number 162414 by HongKing last updated on 29/Dec/21 $$\mathrm{Prove}\:\mathrm{the}\:\mathrm{Identity}\:\mathrm{for}\:\mathrm{any}\:\left(\mathrm{a},\mathrm{n}\right)\:\mathrm{in}\:\mathrm{Real}\:\mathrm{Number} \\ $$$$\left(\mathrm{1}\:+\:\mathrm{a}\right)\centerdot\mathrm{a}^{\left[\boldsymbol{\mathrm{n}}\right]} \:=\:\mathrm{a}\:\centerdot\:\mathrm{a}^{\mathrm{2}\left[\frac{\boldsymbol{\mathrm{n}}}{\mathrm{2}}\right]} \:+\:\mathrm{a}^{\mathrm{2}\left[\frac{\boldsymbol{\mathrm{n}}+\mathrm{1}}{\mathrm{2}}\right]} \\ $$$$\left[\ast\right]\:\mathrm{Greatest}\:\mathrm{Integer}\:\mathrm{Function} \\ $$ Answered by mindispower last updated on 31/Dec/21…