Question Number 226728 by hardmath last updated on 11/Dec/25 $$\mathrm{Find}:\:\:\:\int_{\mathrm{0}} ^{\:\frac{\boldsymbol{\pi}}{\mathrm{4}}} \:\frac{\mathrm{dx}}{\mathrm{1}\:+\:\mathrm{sin}^{\mathrm{2}} \mathrm{x}}\:=\:? \\ $$ Answered by Ghisom_ last updated on 12/Dec/25 $$\mathrm{there}'\mathrm{s}\:\mathrm{an}\:\mathrm{easy}\:\mathrm{but}\:\mathrm{boring}\:\mathrm{solution}\:\mathrm{using} \\ $$$${t}=\mathrm{tan}\:{x}…
Question Number 226713 by ajfour last updated on 11/Dec/25 $$\:\:{If}\: \\ $$$${p}=\frac{\mathrm{1}}{{x}}−\frac{\mathrm{1}}{{x}^{\mathrm{2}} } \\ $$$${what}\:{should}\:{p}=\frac{\left({N}\right)_{\mathrm{4}} }{\left({D}\right)_{\mathrm{4}} } \\ $$$$\left({N}\right)_{\mathrm{4}} \:{means}\:{numerator}\:{of}\:{p}\:\:{in} \\ $$$${quaternary}\:{for}\:{x}\:{to}\:{be}\:\mathrm{777}? \\ $$ Terms…
Question Number 226687 by fantastic2 last updated on 10/Dec/25 $${im}\:{back}\:{guys}! \\ $$$${my}\:{exams}\:{are}\:{over} \\ $$$$ \\ $$ Commented by fantastic2 last updated on 10/Dec/25 $${efgh} \\…
Question Number 226675 by hardmath last updated on 10/Dec/25 Commented by hardmath last updated on 10/Dec/25 $$ \\ $$The image is not fully displayed, the…
Question Number 226603 by Spillover last updated on 07/Dec/25 $${Formulate}\:{the}\:{differential} \\ $$$${equation}\:{of}\:{the}\:{solution} \\ $$$$\left({a}\right){y}={Ae}^{{bx}+\mathrm{1}} \\ $$$$\left({b}\right){y}={A}\mathrm{sin}\:{x}+{B}\mathrm{cos}\:{x} \\ $$$$ \\ $$ Answered by mr W last…
Question Number 226577 by mr W last updated on 06/Dec/25 Answered by Ghisom_ last updated on 06/Dec/25 $${x}_{\mathrm{1}} =\alpha+\sqrt{\beta}+\sqrt{\gamma}+\sqrt{\beta\gamma} \\ $$$${x}_{\mathrm{2}} =\alpha−\sqrt{\beta}−\sqrt{\gamma}+\sqrt{\beta\gamma} \\ $$$${x}_{\mathrm{3}} =\alpha−\sqrt{\beta}+\sqrt{\gamma}−\sqrt{\beta\gamma}…
Question Number 226569 by hardmath last updated on 05/Dec/25 $$\mathrm{a}^{\mathrm{4}} \:+\:\mathrm{b}^{\mathrm{4}} \:+\:\mathrm{c}^{\mathrm{4}} \:=\:\mathrm{2d}^{\mathrm{2}} \\ $$$$\mathrm{Prove}\:\mathrm{that}\:\mathrm{the}\:\mathrm{equation}\:\mathrm{has}\:\mathrm{an}\:\mathrm{infinite} \\ $$$$\mathrm{number}\:\mathrm{of}\:\mathrm{natural}\:\mathrm{solutions} \\ $$ Commented by mr W last updated…
Question Number 226558 by hardmath last updated on 04/Dec/25 Answered by mr W last updated on 04/Dec/25 Commented by hardmath last updated on 05/Dec/25 $$\mathrm{thank}\:\mathrm{you}\:\mathrm{my}\:\mathrm{dear}\:\mathrm{professor}\:\mathrm{cool}…
Question Number 226536 by hardmath last updated on 02/Dec/25 $$\mathrm{If}\:\:\:\left(\mathrm{x}+\left(\mathrm{2a}^{\mathrm{2}} +\mathrm{5}\right)\right)\left(\mathrm{x}−\left(\mathrm{2a}^{\mathrm{2}} +\mathrm{7}\right)\right)\:\leqslant\:\mathrm{0} \\ $$$$\:\:\:\:\:\:\:\mathrm{x}\in\left[−\left(\mathrm{a}^{\mathrm{2}} +\mathrm{8a}−\mathrm{10}\right)\:;\:\left(\mathrm{a}^{\mathrm{2}} +\mathrm{9a}−\mathrm{11}\right)\right] \\ $$$$\mathrm{Find}:\:\boldsymbol{\mathrm{a}}\:=\:? \\ $$ Answered by gregori last updated…
Question Number 226533 by mr W last updated on 02/Dec/25 Answered by mr W last updated on 03/Dec/25 $${cube}\:{roots}\:{of}\:{unit}:\:\mathrm{1},\:\omega,\:\omega^{\mathrm{2}} \\ $$$$\mathrm{1}+\omega+\omega^{\mathrm{2}} =\mathrm{0} \\ $$$${e}^{{x}} =\underset{{n}=\mathrm{0}}…