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Author: Tinku Tara

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Question Number 216710 by universe last updated on 16/Feb/25 $$\:\:\:\mathrm{let}\:{y}_{\mathrm{1}} \:,\:{y}_{\mathrm{2}} \:,\:{y}_{\mathrm{3}\:\:} …\:{y}_{{p}} \:\mathrm{be}\:\mathrm{fixed}\:\mathrm{positive}\:\mathrm{number} \\ $$$$\:\:\:\mathrm{consider}\:\mathrm{the}\:\mathrm{sequences} \\ $$$$\:{s}_{{n}} \:=\:\:\frac{{y}_{\mathrm{1}} ^{{n}} +{y}_{\mathrm{2}} ^{{n}} +{y}_{\mathrm{3}} ^{{n}} +…+{y}_{{p}}…

dz-1-sin-z-cos-z-sec-2-z-dz-sec-2-z-tan-z-multiply-sec-2-z-sec-2-z-1-tan-2-z-sec-2-z-dz-1-tan-z-tan-2-z-s-tan-z-ds-sec-2-z-dz-ds-s-2-s-1-ds-

Question Number 216706 by issac last updated on 16/Feb/25 $$\int\:\:\:\frac{\mathrm{d}{z}}{\mathrm{1}+\mathrm{sin}\left({z}\right)\mathrm{cos}\left({z}\right)}= \\ $$$$\int\:\:\:\frac{\mathrm{sec}^{\mathrm{2}} \left({z}\right)\mathrm{d}{z}}{\mathrm{sec}^{\mathrm{2}} \left({z}\right)+\mathrm{tan}\left({z}\right)}\:\mathrm{multiply}\:\mathrm{sec}^{\mathrm{2}} \left({z}\right) \\ $$$$\mathrm{sec}^{\mathrm{2}} \left({z}\right)=\mathrm{1}+\mathrm{tan}^{\mathrm{2}} \left({z}\right) \\ $$$$\int\:\:\:\frac{\mathrm{sec}^{\mathrm{2}} \left({z}\right)\:\mathrm{d}{z}}{\mathrm{1}+\mathrm{tan}\left({z}\right)+\mathrm{tan}^{\mathrm{2}} \left({z}\right)} \\ $$$${s}=\mathrm{tan}\left({z}\right)…

Question-216670

Question Number 216670 by ahmed2025 last updated on 15/Feb/25 Answered by shunmisaki007 last updated on 15/Feb/25 $$\int\mathrm{cos}^{−{x}} \left(\pi\right){dx}=\int\left(−\mathrm{1}\right)^{−{x}} {dx} \\ $$$$\:\:\:=\int\left({e}^{\pi{i}} \right)^{−{x}} {dx}=\int{e}^{−\pi{ix}} {dx} \\…

Question-216655

Question Number 216655 by abdelsalam last updated on 14/Feb/25 Answered by issac last updated on 14/Feb/25 $${y}^{\mathrm{4}} ={y}^{\mathrm{2}} −{x}^{\mathrm{2}} \\ $$$${y}^{\mathrm{4}} −{y}^{\mathrm{2}} +\frac{\mathrm{1}}{\mathrm{4}}−\frac{\mathrm{1}}{\mathrm{4}}=−{x}^{\mathrm{2}} \\ $$$$\left({y}^{\mathrm{2}}…