Question Number 63645 by mathmax by abdo last updated on 06/Jul/19 $${n}\:{integr}\:{natural}\:{prove}\:{that}\:\mathrm{5}\:{divide}\:{n}^{\mathrm{5}} −{n} \\ $$ Commented by Prithwish sen last updated on 06/Jul/19 $$\mathrm{n}=\mathrm{5k}\:\Rightarrow\mathrm{n}^{\mathrm{5}} −\mathrm{n}\:\mathrm{divided}\:\mathrm{by}\:\mathrm{5}…
Question Number 129181 by liberty last updated on 13/Jan/21 $$\:\mathrm{Given}\:\mathrm{tan}\:^{\mathrm{2}} \mathrm{x}−\mathrm{5tan}\:\mathrm{x}=\mathrm{3}\:\mathrm{has}\:\mathrm{the}\:\mathrm{roots}\: \\ $$$$\mathrm{are}\:\mathrm{x}_{\mathrm{1}} \:\mathrm{and}\:\mathrm{x}_{\mathrm{2}} .\:\mathrm{Find}\:\mid\:\mathrm{cos}\:\mathrm{x}_{\mathrm{1}} .\mathrm{cos}\:\mathrm{x}_{\mathrm{2}} \:\mid\:. \\ $$ Commented by MJS_new last updated on…
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Question Number 63643 by vajpaithegrate@gmail.com last updated on 06/Jul/19 $$\mathrm{P}\left(\alpha,\beta\right)\:\mathrm{Q}\left(\gamma,\delta\right)\:\mathrm{are}\:\mathrm{two}\:\mathrm{points}\:\mathrm{lie}\:\mathrm{on}\:\mathrm{curve} \\ $$$$\mathrm{tan}^{\mathrm{2}} \left(\mathrm{x}+\mathrm{y}\right)+\mathrm{cos}^{\mathrm{2}} \left(\mathrm{x}+\mathrm{y}\right)+\mathrm{y}^{\mathrm{2}} +\mathrm{2y}=\mathrm{0}\:\mathrm{on} \\ $$$$\mathrm{XY}\:\mathrm{plane}.\mathrm{If}\:\mathrm{d}=\mathrm{PQ}\:\mathrm{then}\:\mathrm{cos}\:\mathrm{d}= \\ $$$$\mathrm{ans}:\pm\mathrm{2n}\pi,\mathrm{n}\in\mathrm{N} \\ $$ Answered by MJS last…
Question Number 63642 by bshahid010@gmail.com last updated on 06/Jul/19 Commented by kaivan.ahmadi last updated on 06/Jul/19 $$\left({x}+\mathrm{2}\right)\left({x}^{\mathrm{2}} −\mathrm{2}{x}+\mathrm{4}\right)=\mathrm{0}\Rightarrow \\ $$$${x}=−\mathrm{2}\Rightarrow\gamma=−\mathrm{2}\Rightarrow\gamma^{\mathrm{2}} =\mathrm{4} \\ $$$$\alpha,\beta\:{are}\:{the}\:{roots}\:{of}\:{x}^{\mathrm{2}} −\mathrm{2}{x}+\mathrm{4}=\mathrm{0} \\…
Question Number 63641 by aliesam last updated on 06/Jul/19 Commented by mathmax by abdo last updated on 06/Jul/19 $${let}\:{A}_{{n}} =\:\int_{\mathrm{0}} ^{\mathrm{1}} \:\:\frac{\mid{sin}\left({n}\pi{x}\right)\mid}{{x}^{\mathrm{2}} \:+\mathrm{1}}\:{dx}\:\Rightarrow\:{lim}_{{n}\rightarrow+\infty} \:{A}_{{n}} ={lim}_{{n}\rightarrow+\infty}…
Question Number 129177 by bramlexs22 last updated on 13/Jan/21 $$\:\mathrm{G}\:=\:\int\:\frac{\mathrm{sin}\:^{\mathrm{2}} \mathrm{x}+\mathrm{sin}\:\mathrm{x}}{\mathrm{1}+\mathrm{sin}\:\mathrm{x}+\mathrm{cos}\:\mathrm{x}}\:\mathrm{dx}\:? \\ $$ Answered by liberty last updated on 13/Jan/21 $$\:\mathrm{G}\:=\:\int\:\frac{\mathrm{sin}\:\mathrm{x}\left(\mathrm{sin}\:\mathrm{x}+\mathrm{1}\right)}{\mathrm{sin}\:\mathrm{x}+\mathrm{2cos}\:^{\mathrm{2}} \left(\frac{\mathrm{x}}{\mathrm{2}}\right)}\:\mathrm{dx} \\ $$$$\:\mathrm{G}\:=\:\int\:\frac{\mathrm{2sin}\:\left(\frac{\mathrm{x}}{\mathrm{2}}\right)\mathrm{cos}\:\left(\frac{\mathrm{x}}{\mathrm{2}}\right)\left\{\mathrm{sin}\:\left(\frac{\mathrm{x}}{\mathrm{2}}\right)+\mathrm{cos}\:\left(\frac{\mathrm{x}}{\mathrm{2}}\right)\right\}^{\mathrm{2}} }{\mathrm{2cos}\:\left(\frac{\mathrm{x}}{\mathrm{2}}\right)\left\{\mathrm{sin}\:\left(\frac{\mathrm{x}}{\mathrm{2}}\right)+\mathrm{cos}\:\left(\frac{\mathrm{x}}{\mathrm{2}}\right)\right\}}\mathrm{dx}…
Question Number 129174 by Study last updated on 13/Jan/21 Commented by Study last updated on 13/Jan/21 $${x}=?\:\:\:\:\:\:\:\left[{y}=???\right. \\ $$ Answered by mr W last updated…
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Question Number 129173 by bramlexs22 last updated on 13/Jan/21 $$\:\mathrm{J}\:=\:\int\:\frac{\mathrm{d}\theta}{\mathrm{tan}\:\theta+\mathrm{cot}\:\theta+\mathrm{sec}\:\theta+\mathrm{csc}\:\theta}\:? \\ $$ Answered by liberty last updated on 13/Jan/21 $$\:\mathrm{J}\:=\:\int\frac{\mathrm{d}\theta}{\frac{\mathrm{sin}\:\theta+\mathrm{1}}{\mathrm{cos}\:\theta}+\frac{\mathrm{cos}\:\theta+\mathrm{1}}{\mathrm{sin}\:\theta}}\:=\:\int\:\frac{\mathrm{cos}\:\theta\:\mathrm{sin}\:\theta\:\mathrm{d}\theta}{\mathrm{sin}\:^{\mathrm{2}} \theta+\mathrm{sin}\:\theta+\mathrm{cos}\:^{\mathrm{2}} \theta+\mathrm{cos}\:\theta} \\ $$$$\:\mathrm{J}\:=\:\int\:\frac{\mathrm{sin}\:\theta\:\mathrm{cos}\:\theta\:\mathrm{d}\theta}{\mathrm{1}+\mathrm{sin}\:\theta+\mathrm{cos}\:\theta}\:=\:\int\:\frac{\mathrm{2sin}\:\left(\frac{\theta}{\mathrm{2}}\right)\mathrm{cos}\:\left(\frac{\theta}{\mathrm{2}}\right)\mathrm{cos}\:\theta}{\mathrm{2sin}\:\left(\frac{\theta}{\mathrm{2}}\right)\mathrm{cos}\:\left(\frac{\theta}{\mathrm{2}}\right)+\mathrm{2cos}\:^{\mathrm{2}} \left(\frac{\theta}{\mathrm{2}}\right)}\mathrm{d}\theta…