Question Number 21464 by mondodotto@gmail.com last updated on 24/Sep/17 Answered by myintkhaing last updated on 24/Sep/17 $$\mathrm{y}+\delta\mathrm{y}=\sqrt{\mathrm{tan}\left(\mathrm{x}+\delta\mathrm{x}\right)} \\ $$$$\delta\mathrm{y}=\sqrt{\mathrm{tan}\left(\mathrm{x}+\delta\mathrm{x}\right)}−\sqrt{\mathrm{tanx}} \\ $$$$\:\:\:\:\:=\left(\sqrt{\mathrm{tan}\left(\mathrm{x}+\delta\mathrm{x}\right)}−\sqrt{\mathrm{tanx}}\right)\frac{\sqrt{\mathrm{tan}\left(\mathrm{x}+\delta\mathrm{x}\right)}+\sqrt{\mathrm{tanx}}}{\:\sqrt{\mathrm{tan}\left(\mathrm{x}+\delta\mathrm{x}\right)}+\sqrt{\mathrm{tanx}}} \\ $$$$\:\:\:\:\:=\frac{\mathrm{tan}\left(\mathrm{x}+\delta\mathrm{x}\right)−\mathrm{tanx}}{\:\sqrt{\mathrm{tan}\left(\mathrm{x}+\delta\mathrm{x}\right)}+\sqrt{\mathrm{tanx}}}=\frac{\frac{\mathrm{tanx}+\mathrm{tan}\delta\mathrm{x}}{\mathrm{1}−\mathrm{tanxtan}\delta\mathrm{x}}−\mathrm{tanx}}{\mathrm{tan}\sqrt{\left(\mathrm{x}+\delta\mathrm{x}\right)}+\sqrt{\mathrm{tanx}}} \\ $$$$=\frac{\mathrm{tan}\delta\mathrm{x}\left(\mathrm{1}+\mathrm{tan}^{\mathrm{2}}…
Question Number 152533 by mnjuly1970 last updated on 29/Aug/21 Answered by Kamel last updated on 29/Aug/21 $${I}\overset{{t}=\sqrt{{x}}} {=}\mathrm{2}\int_{\mathrm{0}} ^{+\infty} \frac{{Ln}\left(\mathrm{1}+{t}\right)}{{t}\left(\mathrm{1}+{t}\right)}{dt}=\mathrm{2}\int_{\mathrm{0}} ^{\mathrm{1}} \frac{{Ln}\left(\mathrm{1}+{t}\right)}{{t}\left(\mathrm{1}+{t}\right)}{dt}+\mathrm{2}\int_{\mathrm{0}} ^{\mathrm{1}} \frac{{Ln}\left(\mathrm{1}+{t}\right)−{Ln}\left({t}\right)}{\mathrm{1}+{t}}{dt} \\…
Question Number 86998 by M±th+et£s last updated on 01/Apr/20 $$\int\frac{\mathrm{6}{e}^{{x}} }{{e}^{\mathrm{2}{x}} −\mathrm{1}}\:{dx} \\ $$ Answered by TANMAY PANACEA. last updated on 01/Apr/20 $$\int\frac{\mathrm{6}{d}\left({e}^{{x}} \right)}{\left({e}^{{x}} +\mathrm{1}\right)\left({e}^{{x}}…
Question Number 21463 by dioph last updated on 24/Sep/17 $$\mathrm{Let}\:{A}\:\mathrm{be}\:\mathrm{the}\:\mathrm{collection}\:\mathrm{of}\:\mathrm{functions} \\ $$$${f}\::\:\left[\mathrm{0},\:\mathrm{1}\right]\:\rightarrow\:\mathbb{R}\:\mathrm{which}\:\mathrm{have}\:\mathrm{an}\:\mathrm{infinite} \\ $$$$\mathrm{number}\:\mathrm{of}\:\mathrm{derivatives}.\:\mathrm{Let}\:{A}_{\mathrm{0}} \:\subset\:{A} \\ $$$$\mathrm{be}\:\mathrm{the}\:\mathrm{subcollection}\:\mathrm{of}\:\mathrm{those}\:\mathrm{functions} \\ $$$${f}\:\mathrm{with}\:{f}\left(\mathrm{0}\right)\:=\:\mathrm{0}.\:\mathrm{Define}\:{D}\::\:{A}_{\mathrm{0}} \:\rightarrow\:{A} \\ $$$$\mathrm{by}\:{D}\left({f}\right)\:=\:{df}/{dx}.\:\mathrm{Use}\:\mathrm{the}\:\mathrm{mean}\:\mathrm{value} \\ $$$$\mathrm{theorem}\:\mathrm{to}\:\mathrm{show}\:\mathrm{that}\:{D}\:\mathrm{is}\:\mathrm{injective}. \\…
Question Number 152532 by mondli66 last updated on 29/Aug/21 Answered by amin96 last updated on 29/Aug/21 $$\begin{cases}{\mathrm{0}+{y}+\mathrm{3}{z}=−\mathrm{4}}\\{{x}+\mathrm{2}{y}+{z}=\mathrm{7}}\\{{x}−\mathrm{2}{y}+\mathrm{0}=\mathrm{1}}\end{cases}\:\:\:\Rightarrow\:\:\begin{pmatrix}{\mathrm{1}}&{\mathrm{2}}&{\mathrm{1}}&{\mathrm{7}}\\{\mathrm{1}}&{−\mathrm{2}}&{\mathrm{0}}&{\mathrm{1}}\\{\mathrm{0}}&{\mathrm{1}}&{\mathrm{3}}&{−\mathrm{4}}\end{pmatrix}^{\left(\mathrm{1}\right)−\left(\mathrm{1}\right)} = \\ $$$$=\begin{pmatrix}{\mathrm{1}}&{\mathrm{2}}&{\mathrm{1}}&{\mathrm{7}}\\{\mathrm{0}}&{\mathrm{4}}&{\mathrm{1}}&{\mathrm{6}}\\{\mathrm{0}}&{\mathrm{1}}&{\mathrm{3}}&{−\mathrm{4}}\end{pmatrix}^{\left(\mathrm{3}\right)×\mathrm{4}−\left(\mathrm{2}\right)} =\begin{pmatrix}{\mathrm{1}}&{\mathrm{2}}&{\mathrm{1}}&{\mathrm{7}}\\{\mathrm{0}}&{\mathrm{4}}&{\mathrm{1}}&{\mathrm{6}}\\{\mathrm{0}}&{\mathrm{0}}&{\mathrm{11}}&{−\mathrm{22}}\end{pmatrix} \\ $$$$\begin{cases}{{x}+\mathrm{2}{y}+{z}=\mathrm{7}}\\{\mathrm{4}{y}+{z}=\mathrm{6}}\\{\mathrm{11}{z}=−\mathrm{22}}\end{cases}\:\:\Rightarrow\:{z}=−\mathrm{2}\:\Rightarrow\:{y}=\mathrm{2}\:\:\Rightarrow{x}=\mathrm{5} \\ $$$${answer}\:\:\:\left(\mathrm{5};\:\mathrm{2}\:;\:−\mathrm{2}\right)…
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Question Number 21459 by x² – y²@gmail.com last updated on 24/Sep/17 $$\mathrm{Solve} \\ $$$$\sqrt{{x}\:+\:\mathrm{1}\:−\:\mathrm{4}\sqrt{{x}\:−\:\mathrm{3}}}\:+\:\sqrt{{x}\:+\:\mathrm{22}\:+\:\mathrm{10}\sqrt{{x}\:−\:\mathrm{3}}}\:=\:\mathrm{7} \\ $$ Answered by Tikufly last updated on 27/Sep/17 $$ \\…
Question Number 86995 by M±th+et£s last updated on 01/Apr/20 $$\int_{\mathrm{0}} ^{\pi} \frac{{a}^{{n}} {sin}^{\mathrm{2}} \left({x}\right)+{b}^{{n}} {cos}^{\mathrm{2}} \left({x}\right)}{{a}^{\mathrm{2}{n}} {sin}^{\mathrm{2}} \left({x}\right)+{b}^{\mathrm{2}{n}} {cos}^{\mathrm{2}} \left({x}\right)}{dx}\:;\:{a}>{b} \\ $$ Answered by TANMAY…
Question Number 86993 by Power last updated on 01/Apr/20 Commented by abdomathmax last updated on 01/Apr/20 $${I}\:=\int_{\mathrm{1}} ^{\mathrm{5}} \left[\mathrm{10}{x}\right]{dx}\:\:{vhangement}\:\mathrm{10}{x}\:={t}\:{give} \\ $$$${I}\:=\frac{\mathrm{1}}{\mathrm{10}}\:\int_{\mathrm{10}} ^{\mathrm{50}} \left[{t}\right]{dt}\:=\frac{\mathrm{1}}{\mathrm{10}}\sum_{{k}=\mathrm{10}} ^{\mathrm{49}} \:\int_{{k}}…
Question Number 21453 by Joel577 last updated on 24/Sep/17 $$\mathrm{If}\: \\ $$$$\left(\mathrm{1}\:+\:{n}\right)\mathrm{sin}\:\mathrm{2}\theta\:+\:\left(\mathrm{1}\:−\:{n}\right)\mathrm{cos}\:\mathrm{2}\theta\:=\:\mathrm{1}\:+\:{n} \\ $$$$\mathrm{find}\:\mathrm{tan}\:\mathrm{2}\theta \\ $$ Answered by $@ty@m last updated on 24/Sep/17 $$\left(\mathrm{1}−{n}\right)\mathrm{cos}\:\mathrm{2}\theta=\left(\mathrm{1}+{n}\right)\left(\mathrm{1}−\mathrm{sin}\:\mathrm{2}\theta\right) \\…