Question Number 63682 by Annmuema last updated on 07/Jul/19 $${given}\:{diameter}\:\mathrm{25}{mm} \\ $$$${half}\:{of}\:{the}\:{drill}\:{point}\:{angle}\:=\mathrm{60} \\ $$$${cutting}\:{velocity}=\mathrm{44000}{mm}/{minute} \\ $$$${length}=\mathrm{60}{mm} \\ $$$${feedrate}=\mathrm{0}.\mathrm{25}{mm}/{revolution} \\ $$$${determine}\:{the}\:{time}\:{needed}\:{to}\:{drill}\:{a}\:{through}\:{hole} \\ $$ Terms of Service…
Question Number 63678 by aliesam last updated on 07/Jul/19 Commented by Prithwish sen last updated on 07/Jul/19 $$\mathrm{Putting}\:\mathrm{x}=\mathrm{0}\:\mathrm{we}\:\mathrm{get}\:\mathrm{a}\:\mathrm{solution}. \\ $$$$\mathrm{Now}\:\mathrm{as}\:\mathrm{the}\:\mathrm{left}\:\mathrm{hand}\:\mathrm{side}\:\mathrm{is}\:\mathrm{an}\:\mathrm{increasing}\: \\ $$$$\mathrm{function}\:\mathrm{and}\:\mathrm{the}\:\mathrm{right}\:\mathrm{hand}\:\mathrm{side}\:\mathrm{is}\:\mathrm{a}\:\mathrm{decreasing}\: \\ $$$$\mathrm{function}\:\mathrm{then}\:\mathrm{the}\:\mathrm{equation}\:\mathrm{has}\:\mathrm{only}\:\mathrm{1}\:\mathrm{solution} \\…
Question Number 129212 by bramlexs22 last updated on 13/Jan/21 $$\:\mathrm{M}\:=\:\int\:\sqrt{\mathrm{x}+\sqrt{\mathrm{x}^{\mathrm{2}} +\mathrm{5}}}\:\mathrm{dx}\:? \\ $$ Answered by TheSupreme last updated on 13/Jan/21 $${x}+\sqrt{{x}^{\mathrm{2}} +\mathrm{5}}={t} \\ $$ Answered…
Question Number 129213 by bramlexs22 last updated on 13/Jan/21 $$\:\mathrm{Given}\:\mathrm{f}\left(\mathrm{x}\right)=\frac{\mathrm{2x}−\mathrm{1}}{\mathrm{2x}+\mathrm{1}}\:\mathrm{and}\:\frac{\mathrm{f}\left(\frac{\mathrm{1}}{\mathrm{x}^{\mathrm{2}} }\right)−\mathrm{1}}{\mathrm{f}\left(\frac{\mathrm{1}}{\mathrm{x}^{\mathrm{2}} }\right)+\mathrm{1}}\:=\:\frac{\mathrm{1}}{\mathrm{x}^{\mathrm{2}} } \\ $$$$\mathrm{then}\:\mathrm{x}\:=? \\ $$ Answered by liberty last updated on 13/Jan/21 $$\:\mathrm{f}\left(\frac{\mathrm{1}}{\mathrm{x}^{\mathrm{2}}…
Question Number 129211 by bramlexs22 last updated on 13/Jan/21 $$\:\:\underset{{x}\rightarrow\mathrm{0}} {\mathrm{lim}}\:\frac{\mathrm{1}}{\mathrm{cos}\:\left(\sqrt{\mathrm{x}}\:\right)^{\frac{\mathrm{1}}{\mathrm{x}}} }\:=?\: \\ $$ Answered by liberty last updated on 13/Jan/21 $$\:\mathcal{L}\:=\:\underset{{x}\rightarrow\mathrm{0}} {\mathrm{lim}}\:\frac{\mathrm{1}}{\mathrm{cos}\:\left(\sqrt{\mathrm{x}}\:\right)^{\frac{\mathrm{1}}{\mathrm{x}}} } \\…
Question Number 63674 by Tawa1 last updated on 07/Jul/19 $$\mathrm{Show}\:\mathrm{that}\:\mathrm{the}\:\mathrm{number}\:\:\mathrm{122}^{\mathrm{n}} \:−\:\mathrm{102}^{\mathrm{n}} \:−\:\mathrm{21}^{\mathrm{n}} \:\:\mathrm{is}\:\mathrm{always}\:\mathrm{one}\:\mathrm{less}\:\mathrm{than}\:\mathrm{a} \\ $$$$\:\mathrm{multiple}\:\mathrm{of}\:\:\mathrm{2020}.\:\:\mathrm{For}\:\mathrm{every}\:\mathrm{positive}\:\mathrm{integer}\:\:\mathrm{n}. \\ $$ Commented by Prithwish sen last updated on 07/Jul/19…
Question Number 63667 by mathmax by abdo last updated on 07/Jul/19 $$\left.\mathrm{1}\right)\:{calculate}\:\int_{\mathrm{0}} ^{\mathrm{2}\pi} \:\:\:\frac{{dt}}{{cost}\:+{x}\:{sint}}\:\:\:{wih}\:{x}\:{from}\:{R}. \\ $$$$\left.\mathrm{2}\right)\:{calculate}\:\:\int_{\mathrm{0}} ^{\mathrm{2}\pi} \:\:\frac{{sint}}{\left({cost}\:+{xsint}\right)^{\mathrm{2}} }{dt} \\ $$$$\left.\mathrm{3}\right)\:{find}\left[{the}\:{value}\:{of}\:\int_{\mathrm{0}} ^{\mathrm{2}\pi} \:\:\frac{{dt}}{{cos}\left(\mathrm{2}{t}\right)+\mathrm{2}{sin}\left(\mathrm{2}{t}\right)}\right. \\ $$…
Question Number 63666 by mathmax by abdo last updated on 07/Jul/19 $${calculate}\:\int_{\mathrm{0}} ^{\mathrm{2}\pi} \:\frac{{dx}}{\mathrm{2}{sinx}\:+{cosx}} \\ $$ Commented by MJS last updated on 07/Jul/19 $$=\mathrm{0} \\…
Question Number 63665 by mathmax by abdo last updated on 07/Jul/19 $${find}\:{the}\:{value}\:{of}\:\sum_{{n}=\mathrm{1}} ^{\infty} \:\:\frac{\left(−\mathrm{1}\right)^{{n}} }{{n}^{\mathrm{2}} \left({n}+\mathrm{1}\right)^{\mathrm{3}} } \\ $$ Commented by mathmax by abdo last…
Question Number 63664 by mathmax by abdo last updated on 07/Jul/19 $${let}\:{f}\left({x}\right)=\int_{\mathrm{0}} ^{\infty} \:\:\frac{{t}^{{a}−\mathrm{1}} }{{x}+{t}^{{n}} }\:{dt}\:\:\:{with}\:\mathrm{0}<{a}<\mathrm{1}\:\:{and}\:\:{x}>\mathrm{0}\:{and}\:{n}\geqslant\mathrm{2} \\ $$$$\left.\mathrm{1}\right)\:{determine}\:{a}\:{explicit}\:{form}\:{of}\:{f}\left({x}\right) \\ $$$$\left.\mathrm{2}\right)\:{calculate}\:{g}\left({x}\right)\:=\int_{\mathrm{0}} ^{\infty} \:\:\frac{{t}^{{a}−\mathrm{1}} }{\left({x}+{t}^{{n}} \right)^{\mathrm{2}} }\:{dt}…