Question Number 168799 by MikeH last updated on 17/Apr/22 $$\mathrm{If}\:\mathrm{the}\:\mathrm{function}\:{f}\:\mathrm{is}\:\mathrm{continuous}\:\mathrm{in} \\ $$$$\left[{a},{b}\right]\: \\ $$$$\mathrm{prove}\:\mathrm{that}\: \\ $$$$\:\underset{{x}\rightarrow\infty\:} {\mathrm{lim}}\frac{{b}−{a}}{{n}}\underset{{k}=\mathrm{1}} {\overset{{n}} {\sum}}{f}\left({a}+\frac{{k}\left({b}−{a}\right)}{{n}}\right)=\int_{{a}} ^{{b}} {f}\left({x}\right){dx} \\ $$ Terms of…
Question Number 168755 by MikeH last updated on 17/Apr/22 $$\int\frac{\mathrm{1}}{{x}+\sqrt{\mathrm{1}−{x}}}\:{dx} \\ $$ Commented by safojontoshtemirov last updated on 17/Apr/22 $$\int\frac{\mathrm{1}}{{x}+\sqrt{\mathrm{1}−{x}}}{dx}=\:\:\:\sqrt{\mathrm{1}−{x}}={t}\:\:\:\mathrm{1}−{x}={t}^{\mathrm{2}} \:\:{x}=\mathrm{1}−{t}^{\mathrm{2}} \:{dx}=−\mathrm{2}{tdt} \\ $$$$−\int\frac{\mathrm{2}{t}}{\mathrm{1}−{t}^{\mathrm{2}} +{t}}{dt}=\int\frac{\mathrm{2}{t}}{{t}^{\mathrm{2}}…
Question Number 103168 by mohammad17 last updated on 13/Jul/20 $${The}\:{particular}\:{solution}\:{of}\:{differential}\:{equation}\: \\ $$$${of}\:\frac{{dy}}{{dx}}+\frac{{y}}{{x}}={k}\:{is}\:{y}=\frac{\mathrm{1}}{{x}}+\mathrm{2}{x}\:{thus}\:,{whats}\:{the}\:{value}\:{of}\:{k}\:? \\ $$ Answered by bemath last updated on 13/Jul/20 $${IF}\:{u}\left({x}\right)=\:{e}^{\int\:\frac{{dx}}{{x}}} \:=\:{e}^{\mathrm{ln}\left({x}\right)} \:=\:{x} \\…
Question Number 103149 by bemath last updated on 13/Jul/20 $${using}\:{first}\:{principal}\: \\ $$$${y}\:=\:\mathrm{ln}\:\left(\mathrm{sin}\:\sqrt{{x}}\right)\:\rightarrow{y}'\:=\:? \\ $$ Answered by bobhans last updated on 13/Jul/20 $${y}'\:=\:\underset{\bigtriangleup{x}\rightarrow\mathrm{0}} {\mathrm{lim}}\frac{\mathrm{ln}\left(\mathrm{sin}\:\sqrt{\mathrm{x}+\bigtriangleup{x}}\right)−\mathrm{ln}\left(\mathrm{sin}\:\sqrt{\mathrm{x}}\right)}{\bigtriangleup{x}} \\ $$$${y}'=\underset{\bigtriangleup{x}\rightarrow\mathrm{0}}…
Question Number 37363 by math khazana by abdo last updated on 12/Jun/18 $${solve}\:{y}^{'} \:\:+{xe}^{−{x}^{\mathrm{2}} } {y}\:\:={e}^{−{x}} \:\:. \\ $$ Answered by tanmay.chaudhury50@gmail.com last updated on…
Question Number 168428 by mnjuly1970 last updated on 10/Apr/22 $$ \\ $$$$\:\:\:\:\:{solve} \\ $$$$\:\:\:\:\:{f}\left({x}\right)=\:{x}\:+\sqrt{{x}^{\:\mathrm{2}} −\mathrm{3}{x}\:+\mathrm{2}} \\ $$$$\:\:\:\:\:\:\:\:{R}\:_{{f}} \:=?\:\:\:\:\:\:\:{R}_{\:{f}} :\:{rang}\:{of}\:\:\:\:{f} \\ $$ Answered by MJS_new last…
Question Number 37344 by math khazana by abdo last updated on 12/Jun/18 $${solve}\:{the}\:{d}.{e}.\:{y}^{'} \:−{xy}\:\:={cosx}\:. \\ $$ Commented by math khazana by abdo last updated on…
Question Number 168348 by mnjuly1970 last updated on 08/Apr/22 Answered by mathman1234 last updated on 09/Apr/22 $$\mathrm{plz}\:\mathrm{help} \\ $$$$\mathrm{if}\:\mathrm{0}<\mathrm{a}<\mathrm{b}\:\mathrm{and}\:\mathrm{x}>\mathrm{0}\:\mathrm{proove}\:\mathrm{that} \\ $$$$\frac{\mathrm{2}}{\pi}\left(\mathrm{1}−\frac{\mathrm{a}}{\mathrm{b}}\right)\leqslant\mathrm{sup}\mid\frac{\mathrm{sin}\left(\mathrm{ax}\right)}{\mathrm{ax}}\:−\:\frac{\mathrm{sin}\left(\mathrm{ax}\right)}{\mathrm{ax}}\mid\leqslant\mathrm{4}\left(\mathrm{1}−\frac{\mathrm{a}}{\mathrm{b}}\right) \\ $$ Commented by…
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Question Number 102804 by bramlex last updated on 11/Jul/20 $$\mathrm{The}\:\mathrm{coordinates}\:\mathrm{of}\:\mathrm{two}\:\mathrm{points}\:\mathrm{A} \\ $$$$\&\:\mathrm{B}\:\mathrm{are}\:\left(\mathrm{0},\mathrm{8}\right)\:\mathrm{and}\:\left(\mathrm{9},\mathrm{4}\right) \\ $$$$\mathrm{respectively}.\:\mathrm{The}\:\mathrm{point}\:\mathrm{P}\:\mathrm{with} \\ $$$$\mathrm{coordinate}\:\left(\mathrm{p},\mathrm{0}\right)\:\mathrm{lies}\:\mathrm{on}\:\mathrm{the} \\ $$$$\mathrm{x}−\mathrm{axis}\:\mathrm{where}\:\mathrm{0}<\mathrm{p}<\mathrm{9}.\:\mathrm{Let}\:\mathrm{s} \\ $$$$\mathrm{denotes}\:\mathrm{the}\:\mathrm{sum}\:\mathrm{of}\:\mathrm{the}\:\mathrm{length}\:\mathrm{of} \\ $$$$\mathrm{two}\:\mathrm{segments}\:\mathrm{PA}\:\mathrm{and}\:\mathrm{PB}\:.\:\mathrm{by} \\ $$$$\mathrm{expressing}\:\mathrm{s}\:\mathrm{in}\:\mathrm{terms}\:\mathrm{of}\:\mathrm{p} \\…