Question Number 21423 by Tinkutara last updated on 23/Sep/17 $$\mathrm{Prove}\:\mathrm{that}\:{n}^{\mathrm{4}} \:+\:\mathrm{4}^{{n}} \:\mathrm{is}\:\mathrm{composite}\:\mathrm{for}\:\mathrm{all} \\ $$$$\mathrm{integer}\:\mathrm{values}\:\mathrm{of}\:{n}\:\mathrm{greater}\:\mathrm{than}\:\mathrm{1}. \\ $$ Answered by dioph last updated on 24/Sep/17 $$\mathrm{If}\:{n}\:\mathrm{is}\:\mathrm{even},\:\mathrm{both}\:{n}^{\mathrm{4}} \:\mathrm{and}\:\mathrm{4}^{{n}}…
Question Number 152492 by nadovic last updated on 28/Aug/21 $$\:\mathrm{A}\:\mathrm{particle}\:\mathrm{is}\:\mathrm{projected}\:\mathrm{upwards}\:\mathrm{with} \\ $$$$\:\mathrm{a}\:\mathrm{velocity}\:\mathrm{of}\:\:\mathrm{96}{ms}^{−\mathrm{1}} .\:\mathrm{In}\:\mathrm{addition}\:\mathrm{to} \\ $$$$\:\mathrm{being}\:\mathrm{subject}\:\mathrm{to}\:\mathrm{gravity},\:\mathrm{it}\:\mathrm{is}\:\mathrm{acted}\:\mathrm{on} \\ $$$$\:\mathrm{by}\:\mathrm{a}\:\mathrm{retardation}\:\mathrm{of}\:\mathrm{16}{t},\:\mathrm{where}\:{t}\:\mathrm{is}\:\mathrm{the} \\ $$$$\:\mathrm{time}\:\mathrm{from}\:\mathrm{the}\:\mathrm{start}\:\mathrm{of}\:\mathrm{the}\:\mathrm{motion}. \\ $$$$\:\mathrm{What}\:\mathrm{is}\:\mathrm{the}\:\mathrm{greatest}\:\mathrm{height}\:\mathrm{attained} \\ $$$$\:\mathrm{by}\:\mathrm{the}\:\mathrm{particle}? \\ $$…
Question Number 21422 by Tinkutara last updated on 23/Sep/17 $$\mathrm{Find}\:\mathrm{all}\:\mathrm{integer}\:\mathrm{values}\:\mathrm{of}\:{a}\:\mathrm{such}\:\mathrm{that} \\ $$$$\mathrm{the}\:\mathrm{quadratic}\:\mathrm{expression} \\ $$$$\left({x}\:+\:{a}\right)\left({x}\:+\:\mathrm{1991}\right)\:+\:\mathrm{1}\:\mathrm{can}\:\mathrm{be}\:\mathrm{factored} \\ $$$$\mathrm{as}\:\mathrm{a}\:\mathrm{product}\:\left({x}\:+\:{b}\right)\left({x}\:+\:{c}\right)\:\mathrm{where}\:{b}\:\mathrm{and} \\ $$$${c}\:\mathrm{are}\:\mathrm{integers}. \\ $$ Commented by Tikufly last updated…
Question Number 86956 by abdomathmax last updated on 01/Apr/20 $${find}\:\int\left(\mathrm{1}−\frac{\mathrm{1}}{{x}^{\mathrm{2}} }\right){arctan}\left(\mathrm{2}{x}\right){dx} \\ $$ Commented by Ar Brandon last updated on 01/Apr/20 $${Let}\:{u}={arctan}\left(\mathrm{2}{x}\right)\:\Rightarrow\:{du}=\mathrm{2}\centerdot\frac{\mathrm{1}}{\mathrm{1}+\left(\mathrm{2}{x}\right)^{\mathrm{2}} }{dx} \\ $$$$\:\:\:\:\:\:\:\:{dv}=\left(\mathrm{1}−\frac{\mathrm{1}}{{x}^{\mathrm{2}}…
Question Number 152494 by mnjuly1970 last updated on 28/Aug/21 $$ \\ $$$$\:\:\:\:{prove}\:{that}\::: \\ $$$$\:\:\:\: \\ $$$$\:\:\:\:\:\:\:\:\:\:\Omega\::=\:\int_{\mathrm{0}\:} ^{\:\infty} \frac{\:\:{e}^{\:−{x}} .\mathrm{ln}\:\left(\frac{\:\mathrm{1}}{\:{x}}\:\right)\:{sin}\:\left(\:{x}\:\right)}{{x}\:}\:{dx}\:=\:\frac{\:\pi}{\:\mathrm{8}}\:\left(\:\mathrm{2}\:\gamma\:+\mathrm{ln}\:\left(\mathrm{2}\:\right)\:\right)\:…\blacksquare\:\:\:\:\:\:\:\:\:\:\: \\ $$$$\:\:\:\:\:\:{m}.{n} \\ $$$$ \\ $$…
Question Number 86957 by abdomathmax last updated on 01/Apr/20 $${find}\:\int\:\:{arctan}\left(\frac{\mathrm{1}−{u}}{\mathrm{1}+{u}}\right){du} \\ $$ Commented by Ar Brandon last updated on 01/Apr/20 $${I}=\int{arctan}\left(\frac{\mathrm{1}−{x}}{\mathrm{1}+{x}}\right){dx} \\ $$$${Let}\:{u}={arctan}\left(\frac{\mathrm{1}−{x}}{\mathrm{1}+{x}}\right)\:\Leftrightarrow\:{u}={arctan}\left({t}\right)\:\:{with}\:\:{t}=\frac{\mathrm{1}−{x}}{\mathrm{1}+{x}} \\ $$$$\Rightarrow\frac{{du}}{{dx}}=\frac{{du}}{{dt}}×\frac{{dt}}{{du}}…
Question Number 86955 by abdomathmax last updated on 01/Apr/20 $${find}\:\int\:\:\frac{{arctan}\left({x}\right)}{{x}}{dx} \\ $$ Commented by Ar Brandon last updated on 01/Apr/20 $${Let}\:{f}\left({a}\right)=\int\frac{{arctan}\left({ax}\right)}{{x}}{dx} \\ $$$$\Rightarrow{f}\:'\:\left({a}\right)=\int\frac{{x}}{{x}}\centerdot\frac{\mathrm{1}}{\mathrm{1}+\left({ax}\right)^{\mathrm{2}} }{dx}=\int\frac{\mathrm{1}}{\mathrm{1}+\left({ax}\right)^{\mathrm{2}} }{dx}…
Question Number 86948 by gny last updated on 01/Apr/20 $$\frac{−{sin}\alpha+\mathrm{3}{cos}\alpha}{\mathrm{4}{cos}\alpha+\mathrm{3}{sin}\alpha} \\ $$$$ \\ $$$${tg}\alpha=−\mathrm{3} \\ $$$${pls}\:{help}\:{me}\:{sir} \\ $$ Commented by Prithwish Sen 1 last updated…
Question Number 152486 by Tawa11 last updated on 28/Aug/21 Answered by mr W last updated on 29/Aug/21 Commented by mr W last updated on 29/Aug/21…
Question Number 21412 by Tinkutara last updated on 23/Sep/17 $$\mathrm{The}\:\mathrm{atomic}\:\mathrm{masses}\:\mathrm{of}\:'\mathrm{He}'\:\mathrm{and}\:'\mathrm{Ne}'\:\mathrm{are} \\ $$$$\mathrm{4}\:\mathrm{and}\:\mathrm{20}\:\mathrm{a}.\mathrm{m}.\mathrm{u}.,\:\mathrm{respectively}.\:\mathrm{The} \\ $$$$\mathrm{value}\:\mathrm{of}\:\mathrm{the}\:\mathrm{de}\:\mathrm{Broglie}\:\mathrm{wavelength}\:\mathrm{of} \\ $$$$'\mathrm{He}'\:\mathrm{gas}\:\mathrm{at}\:−\mathrm{73}°\mathrm{C}\:\mathrm{is}\:“\mathrm{M}''\:\mathrm{times}\:\mathrm{that}\:\mathrm{of} \\ $$$$\mathrm{the}\:\mathrm{de}\:\mathrm{Broglie}\:\mathrm{wavelength}\:\mathrm{of}\:'\mathrm{Ne}'\:\mathrm{at} \\ $$$$\mathrm{727}°\mathrm{C}\:'\mathrm{M}'\:\mathrm{is} \\ $$ Terms of Service…