Question Number 82531 by Power last updated on 22/Feb/20 Commented by abdomathmax last updated on 24/Feb/20 $${I}\:=\int\:\frac{\sqrt{\mathrm{1}−{x}}}{{x}}{dx}\:+\int\:\frac{\sqrt{\mathrm{2}−{x}^{\mathrm{2}} }}{{x}}{dx}\:+\int\:\:\frac{\sqrt{\mathrm{4}−{x}^{\mathrm{4}} }}{{x}}{dx}\:{we}\:{have} \\ $$$$\int\:\:\frac{\sqrt{\mathrm{1}−{x}}}{{x}}{dx}\:\:{changement}\:\sqrt{\mathrm{1}−{x}}={t}\:{give}\mathrm{1}−{x}={t}^{\mathrm{2}} \:\Rightarrow \\ $$$${x}=\mathrm{1}−{t}^{\mathrm{2}} \:\Rightarrow\int\:\:\frac{\sqrt{\mathrm{1}−{x}}}{{x}}{dx}\:=\:\int\:\:\frac{{t}}{\mathrm{1}−{t}^{\mathrm{2}}…
Question Number 82528 by TawaTawa last updated on 22/Feb/20 Commented by TawaTawa last updated on 22/Feb/20 $$\mathrm{Please}\:\mathrm{help}. \\ $$ Terms of Service Privacy Policy Contact:…
Question Number 16990 by 1234Hello last updated on 29/Jun/17 $$\mathrm{If}\:\mathrm{for}\:\mathrm{positive}\:\mathrm{integers}\:{a}\:\mathrm{and}\:{b}, \\ $$$${a}\:+\:{b}\:=\:\frac{{a}}{{b}}\:+\:\frac{{b}}{{a}},\:\mathrm{find}\:{a}^{\mathrm{2}} \:+\:{b}^{\mathrm{2}} . \\ $$ Commented by RasheedSoomro last updated on 29/Jun/17 $${a}\:+\:{b}\:=\:\frac{{a}}{{b}}\:+\:\frac{{b}}{{a}} \\…
Question Number 82520 by peter frank last updated on 22/Feb/20 Commented by mathmax by abdo last updated on 22/Feb/20 $${let}\:{S}_{{n}} =\sum_{{k}=\mathrm{0}} ^{{n}} \:\mathrm{2}^{{k}} \:{cos}\left(\mathrm{2}{k}\theta\right)\:\Rightarrow{S}_{{n}} ={Re}\left(\sum_{{k}=\mathrm{0}}…
Question Number 82521 by jagoll last updated on 22/Feb/20 $${given}\:\frac{{dy}}{{dx}}\:=\:{xy}^{\mathrm{3}} \\ $$$${Determine}\:{the}\:{concavity}\:{of}\:{all} \\ $$$${solution}\:{curves}\:{for}\:{the}\:{given}\: \\ $$$${differential}\:{equation}\:{in}\: \\ $$$${quadrant}\:{iv}.\: \\ $$ Commented by jagoll last updated…
Question Number 82519 by peter frank last updated on 22/Feb/20 Commented by mathmax by abdo last updated on 23/Feb/20 $${A}_{\theta} =\sum_{{n}=\mathrm{0}} ^{\infty} \:\frac{\left(−\mathrm{1}\right)^{{n}} }{\mathrm{2}^{{n}} }{cos}\left({n}\theta\right)\:\:=\sum_{{n}=\mathrm{0}}…
Question Number 82517 by naka3546 last updated on 22/Feb/20 $$\underset{\:\mathrm{1}} {\int}\overset{\:\infty} {\:}\:\frac{\mathrm{ln}\:{x}}{{x}^{\mathrm{2}} \:+\:\mathrm{1}}\:{dx}\:\:,\:\:\:{Convergent}\:\:{or}\:\:{Divergent}\:\:? \\ $$ Commented by Tony Lin last updated on 22/Feb/20 $$\int_{\mathrm{1}} ^{\infty}…
Question Number 16980 by mrW1 last updated on 29/Jun/17 $$\mathrm{To}\:\mathrm{Q16066}: \\ $$$$\mathrm{I}\:\mathrm{have}\:\mathrm{posted}\:\mathrm{my}\:\mathrm{solution}\:\mathrm{there}. \\ $$$$\mathrm{Those}\:\mathrm{who}\:\mathrm{are}\:\mathrm{intetested}\:\mathrm{in}\:\mathrm{this}\:\mathrm{interesting} \\ $$$$\mathrm{question}\:\mathrm{please}\:\mathrm{have}\:\mathrm{a}\:\mathrm{critical}\:\mathrm{view}\:\mathrm{at} \\ $$$$\mathrm{it}.\:\mathrm{Maybe}\:\mathrm{there}\:\mathrm{are}\:\mathrm{alternative}\:\mathrm{solutions} \\ $$$$\mathrm{which}\:\mathrm{are}\:\mathrm{easier}\:\mathrm{and}\:\mathrm{more}\:\mathrm{direct}\:\mathrm{and} \\ $$$$\mathrm{straight}\:\mathrm{on}. \\ $$ Commented…
Question Number 82514 by naka3546 last updated on 22/Feb/20 $${Find}\:\:\:{maximum}\:\:{of}\:\:{n}\:\:{such}\:\:{that}\:\: \\ $$$${n}^{\mathrm{2}} \:+\:{n}\:+\:\mathrm{1}\:\:{is}\:\:{factor}\:\:{of}\:\:\mathrm{19}{n}^{\mathrm{2019}} \:+\:\mathrm{1}\:. \\ $$ Commented by naka3546 last updated on 22/Feb/20 $${n}\:\:\in\:\:\mathbb{N} \\…
Question Number 148048 by puissant last updated on 25/Jul/21 $$\mathrm{2}^{\mathrm{x}} −\mathrm{3}^{\left(\frac{\mathrm{x}}{\mathrm{2}}\right)} =\mathrm{1} \\ $$ Answered by mindispower last updated on 25/Jul/21 $$\mathrm{2}^{{x}} −\left(\sqrt{\mathrm{3}}\right)^{{x}} =\mathrm{1} \\…