Question Number 64973 by Tawa1 last updated on 23/Jul/19 Answered by mr W last updated on 23/Jul/19 $$\left(\mathrm{1}\right) \\ $$$$\frac{{dy}}{{dx}}={u} \\ $$$$\frac{{d}^{\mathrm{2}} {y}}{{dx}^{\mathrm{2}} }=\frac{{du}}{{dx}}=\frac{{du}}{{dy}}×\frac{{dy}}{{dx}}={u}\frac{{du}}{{dy}} \\…
Question Number 64971 by Tawa1 last updated on 23/Jul/19 Commented by Tony Lin last updated on 23/Jul/19 $$\frac{\frac{{x}}{\mathrm{2}}}{{sin}\theta}=\frac{\frac{{x}}{\mathrm{2}}+\mathrm{1}}{{sin}\left(\mathrm{90}°+\theta\right)}=\frac{{x}}{{sin}\left(\mathrm{90}°−\mathrm{2}\theta\right)} \\ $$$$\Rightarrow\frac{\frac{{x}}{\mathrm{2}}}{{sin}\theta}=\frac{\frac{{x}}{\mathrm{2}}+\mathrm{1}}{{cos}\theta}=\frac{{x}}{{cos}\mathrm{2}\theta} \\ $$$${let}\:{cos}\theta={t} \\ $$$$\Rightarrow\frac{\frac{{x}}{\mathrm{2}}}{\:\sqrt{\mathrm{1}−{t}^{\mathrm{2}} }}=\frac{\frac{{x}}{\mathrm{2}}+\mathrm{1}}{{t}}=\frac{{x}}{\mathrm{2}{t}^{\mathrm{2}}…
Question Number 64970 by mathmax by abdo last updated on 23/Jul/19 $${let}\:{f}\left({a}\right)=\int_{\mathrm{0}} ^{\infty} \:\:\frac{{cos}\left({x}^{\mathrm{2}} \right)\:+{sin}\left({x}^{\mathrm{2}} \right)}{\left({x}^{\mathrm{2}} \:+{a}^{\mathrm{2}} \right)^{\mathrm{2}} }\:{dx}\:\:\:{with}\:{a}>\mathrm{0} \\ $$$$\left.\mathrm{1}\right)\:{calculate}\:{f}\left({a}\right) \\ $$$$\left.\mathrm{2}\right)\:{find}\:{the}\:{values}\:{of}\:\int_{\mathrm{0}} ^{\infty} \:\:\frac{{cos}\left({x}^{\mathrm{2}}…
Question Number 64968 by ajfour last updated on 23/Jul/19 Commented by ajfour last updated on 25/Jul/19 $${Instead}\:{let}\:\:{y}={Ax}^{\mathrm{2}} \:,\:{find}\:{side} \\ $$$${length}\:{of}\:{such}\:{a}\:{rhombus}\:{that} \\ $$$${fits}\:{in},\:{the}\:{way}\:{shown}.\:{Also} \\ $$$${find}\:{A}.\:\:\:\:\:\:\: \\…
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Question Number 130496 by Adel last updated on 26/Jan/21 $$\frac{\mathrm{d}}{\mathrm{dx}}\left(\mathrm{x}!\right)=? \\ $$ Answered by MJS_new last updated on 26/Jan/21 $${x}!\:\mathrm{is}\:\mathrm{defined}\:\mathrm{for}\:{x}\in\mathbb{N}\:\Rightarrow\:\mathrm{no}\:\mathrm{derivate}\:\mathrm{exists} \\ $$$$ \\ $$$$\mathrm{if}\:\mathrm{you}\:\mathrm{mean}\:{x}!=\Gamma\:\left({x}+\mathrm{1}\right) \\…
Question Number 64955 by Tony Lin last updated on 23/Jul/19 $${prove}\:{that}\:\underset{{k}=\mathrm{1}} {\overset{{n}} {\sum}}{k}\centerdot\frac{\left({k}+\mathrm{1}\right)!}{\mathrm{2}^{{k}+\mathrm{1}} }=\frac{\left({n}+\mathrm{2}\right)!}{\mathrm{2}^{{n}+\mathrm{1}} }−\mathrm{1} \\ $$ Commented by ~ À ® @ 237 ~…
Question Number 130486 by benjo_mathlover last updated on 26/Jan/21 $$\:\int_{\mathrm{0}} ^{\:{x}} \:\frac{\mathrm{cos}\:{t}\:\sqrt[{\mathrm{4}}]{\mathrm{sin}^{\mathrm{3}} \:{t}}}{\left(\mathrm{sin}\:{x}−\mathrm{sin}\:{t}\right)^{\mathrm{3}/\mathrm{4}} }\:{dt}\:? \\ $$ Answered by MJS_new last updated on 26/Jan/21 $$\int\mathrm{cos}\:{t}\:\left(\frac{\mathrm{sin}\:{t}}{\mathrm{sin}\:{x}\:−\mathrm{sin}\:{t}}\right)^{\mathrm{3}/\mathrm{4}} {dt}=…
Question Number 64951 by naka3546 last updated on 23/Jul/19 Answered by mr W last updated on 23/Jul/19 Commented by mr W last updated on 23/Jul/19…