Question Number 129068 by bagjagunawan last updated on 12/Jan/21 Answered by mnjuly1970 last updated on 12/Jan/21 $$\:\:{ans}\:::\:\:\pi{ln}\left(\mathrm{1}−{e}^{−\pi{n}} \right)\:…. \\ $$ Terms of Service Privacy Policy…
Question Number 63532 by Rio Michael last updated on 05/Jul/19 $${prove}\:{that}\:{there}\:{exist}\:{unique}\:{intergers}\:{p}\:{and}\:{s}\:{sucb}\:{that} \\ $$$${a}\:=\:{bp}\:+\:{s}\:{with}\:−\frac{\mid{b}\mid}{\mathrm{2}}<\:{s}\:\leqslant\frac{\mid{b}\mid}{\mathrm{2}} \\ $$$${hence}\:{find}\:{p}\:{and}\:{s}\:{given}\:{that}\:{a}=\mathrm{49}\:{and}\:{b}=\mathrm{26} \\ $$ Answered by MJS last updated on 05/Jul/19 $${a}={bp}+{s}\:\Rightarrow\:{s}={a}−{bp}…
Question Number 129064 by Eric002 last updated on 12/Jan/21 $${valuate}\:{the}\:{following}\:{integral} \\ $$$${I}=\int_{\mathrm{1}} ^{\infty} \frac{{dt}}{\left({t}\right)^{\mathrm{2}{k}+{v}+\frac{\mathrm{3}}{\mathrm{2}}} \sqrt{{t}−\mathrm{1}}} \\ $$$${and}\:{prove}\:{that}: \\ $$$${I}=\sqrt{\pi}\frac{\Gamma\left({v}+\mathrm{1}\right)}{\Gamma\left({v}+\frac{\mathrm{3}}{\mathrm{2}}\right)}\:\left(\frac{\left(\frac{{v}+\mathrm{1}}{\mathrm{2}}\right)_{{k}} \left(\mathrm{1}+\frac{{v}}{\mathrm{2}}\right)_{{k}} }{\left(\frac{{v}}{\mathrm{2}}+\frac{\mathrm{3}}{\mathrm{4}}\right)_{{k}} \left(\frac{{v}}{\mathrm{2}}+\frac{\mathrm{5}}{\mathrm{4}}\right)_{{k}} }\right) \\ $$…
Question Number 129063 by benjo_mathlover last updated on 12/Jan/21 $$\:\mathrm{If}\:\mathrm{sin}\:\mathrm{x}−\mathrm{cos}\:\mathrm{x}\:=\:\frac{\mathrm{1}}{\mathrm{5}} \\ $$$$\:\mathrm{then}\:\begin{cases}{\mathrm{sin}\:\mathrm{x}=?}\\{\mathrm{cos}\:\mathrm{x}=?}\end{cases} \\ $$ Answered by liberty last updated on 12/Jan/21 $$\:\left(\mathrm{sin}\:\mathrm{x}−\mathrm{cos}\:\mathrm{x}\right)^{\mathrm{2}} =\frac{\mathrm{1}}{\mathrm{25}}\:\Rightarrow\mathrm{1}−\mathrm{2sin}\:\mathrm{x}\:\mathrm{cos}\:\mathrm{x}\:=\:\frac{\mathrm{1}}{\mathrm{25}} \\ $$$$\:\mathrm{let}\:\begin{cases}{{p}=\mathrm{5sin}\:{x}}\\{{q}=\mathrm{5cos}\:{x}}\end{cases}\:{then}\:\mathrm{we}\:\mathrm{have}\:\begin{cases}{{p}−{q}=\mathrm{1}}\\{{pq}=\mathrm{12}}\end{cases}…
Question Number 129060 by benjo_mathlover last updated on 12/Jan/21 $$\:\phi\:=\:\int\:\frac{\mathrm{ln}\:\left(\mathrm{x}\right)}{\mathrm{x}^{\mathrm{2}} }\:\mathrm{dx}\: \\ $$ Answered by liberty last updated on 12/Jan/21 $$\:\mathrm{let}\:\mathrm{ln}\:\left(\mathrm{x}\right)=\mathrm{h}\:\Rightarrow\mathrm{x}\:=\:\mathrm{e}^{\mathrm{h}} \\ $$$$\:\phi\:=\:\int\:\frac{\mathrm{h}}{\mathrm{e}^{\mathrm{2h}} }\:\left(\mathrm{e}^{\mathrm{h}} \:\mathrm{dh}\:\right)=\:\int\:\mathrm{h}.\mathrm{e}^{−\mathrm{h}}…
Question Number 129061 by behi83417@gmail.com last updated on 12/Jan/21 Commented by mr W last updated on 12/Jan/21 $${is}\:{it}\:{not}\:{obvious}\:{that}\:{the}\:{maximal} \\ $$$${equilateral}\:{has}\:{the}\:{side}\:{length}\: \\ $$$$\mathrm{1}/\mathrm{cos}\:\mathrm{15}°=\sqrt{\mathrm{6}}−\sqrt{\mathrm{2}}? \\ $$ Commented…
Question Number 63522 by Tawa1 last updated on 05/Jul/19 Commented by Tawa1 last updated on 05/Jul/19 Find the real parameter “m” such that cross cutting of mx + 2y - 1 = 0 and 2x + my + 3 = 0 give slopes equation belongs x - y - 3 = 0 Commented by MJS last updated on 05/Jul/19 $$\left(\mathrm{1}\right)\:{mx}+\mathrm{2}{y}−\mathrm{1}=\mathrm{0}…
Question Number 129057 by pipin last updated on 12/Jan/21 $$\: \\ $$$$\:\int_{\mathrm{1}} ^{\infty} \frac{\mathrm{1}}{\mathrm{1}+\mathrm{x}^{\mathrm{3}} }\mathrm{dx}\:=\:… \\ $$ Answered by MJS_new last updated on 12/Jan/21 $$\int\frac{{dx}}{{x}^{\mathrm{3}}…
Question Number 63519 by Rio Michael last updated on 05/Jul/19 $${consider}\:{the}\:{general}\:{definite}\:{intergral}\:\: \\ $$$$\:{I}_{{n}} =\int_{\mathrm{0}} ^{\frac{\pi}{\mathrm{2}}} {sin}^{{n}} {xdx} \\ $$$$\left.{a}\right)\:{prove}\:{that}\:{for}\:{n}\geqslant\mathrm{2},\:{nI}_{{n}} =\left({n}−\mathrm{1}\right){I}_{{n}−\mathrm{2}} . \\ $$$$\left.{b}\left.\right)\left.\:{Find}\:{the}\:{values}\:{of}\:\:\boldsymbol{{i}}\right)\int_{\mathrm{0}} ^{\frac{\pi}{\mathrm{2}}} {sin}^{\mathrm{5}}…
Question Number 129052 by oustmuchiya@gmail.com last updated on 12/Jan/21 $${Let}\:\boldsymbol{\mathrm{p}}\:{and}\:\boldsymbol{\mathrm{Q}}\:{be}\:{points}\:{on}\:{the}\:{curve} \\ $$$$\boldsymbol{{y}}=\boldsymbol{{x}}^{\mathrm{2}} −\mathrm{2}\boldsymbol{{x}}\:{while}\:\boldsymbol{{x}}=\mathrm{2}\:{and}\:\boldsymbol{{x}}=\mathrm{2}+\boldsymbol{{h}} \\ $$$$\boldsymbol{{respectively}}.\:\boldsymbol{{E}}{press}\:{the}\:{gradient} \\ $$$${of}\:\boldsymbol{\mathrm{P}}{Q}\:{in}\:{terms}\:{of}\:\boldsymbol{{h}}. \\ $$ Commented by benjo_mathlover last updated on…