Question Number 213660 by mnjuly1970 last updated on 13/Nov/24 $$ \\ $$$$\:\:\:\:\:{prove}\:{that}\:… \\ $$$$\mathrm{lim}_{{n}\rightarrow\infty} \int_{\mathrm{0}} ^{\:\mathrm{3}} \frac{\:{x}^{\mathrm{2}} \:\left(\mathrm{1}−{x}\:\right){x}^{{n}} \:}{\mathrm{1}+\:{x}^{\mathrm{2}{n}} }\:{dx}\overset{?} {=}\mathrm{0} \\ $$$$\:\:\:\:\:−−−−−−−−−−− \\ $$$$…
Question Number 213662 by Spillover last updated on 13/Nov/24 Answered by mathmax last updated on 14/Nov/24 $${I}=\int_{\mathrm{0}} ^{\infty} \frac{{x}^{\mathrm{2}} }{\mathrm{1}+{e}^{{x}} }{dx}\:\Rightarrow{I}=\int_{\mathrm{0}} ^{\infty} \frac{{e}^{−{x}} {x}^{\mathrm{2}} }{\mathrm{1}+{e}^{−{x}}…
Question Number 213659 by efronzo1 last updated on 13/Nov/24 $$\:\:\:\:\underset{−\infty} {\overset{\infty} {\int}}\:\frac{\mid\mathrm{24x}−\mathrm{24}\mid−\mathrm{20}}{\mathrm{22}^{\mathrm{x}} +\mathrm{22}}\:\mathrm{dx}\:=? \\ $$ Answered by MrGaster last updated on 03/Feb/25 $$=\frac{\mathrm{40}\:\mathrm{ln}\:\mathrm{22}}{\left(\mathrm{ln}\:\mathrm{22}\right)^{\mathrm{2}} } \\…
Question Number 213656 by efronzo1 last updated on 12/Nov/24 Answered by golsendro last updated on 13/Nov/24 $$\:\:\left(\mathrm{i}\right)\:\mathrm{g}\left(\mathrm{4}−\mathrm{x}\right)=\:−\mathrm{g}\left(\mathrm{x}\right)\: \\ $$$$\:\:\:\:\:\:\:\:\underset{\mathrm{0}} {\overset{\mathrm{4}} {\int}}\:\mathrm{g}\left(\mathrm{x}\right)\mathrm{dx}\:=\:\underset{\mathrm{0}} {\overset{\mathrm{4}} {\int}}\:\mathrm{g}\left(\mathrm{4}−\mathrm{x}\right)\mathrm{dx}\: \\ $$$$\:\:\:\:\:\:\:\:\underset{\mathrm{0}}…
Question Number 213652 by MathematicalUser2357 last updated on 12/Nov/24 $$\mathrm{Show}\:\mathrm{that}\:\int{xdx}=\frac{{x}^{\mathrm{2}} }{{x}}+{C}. \\ $$ Answered by MathematicalUser2357 last updated on 12/Nov/24 $$ \\ $$ Terms of…
Question Number 213628 by Thomaseinstein last updated on 10/Nov/24 Answered by MrGaster last updated on 03/Feb/25 $$\left.=\frac{\mathrm{1}}{\mathrm{2}}\left({x}^{\mathrm{4}} \mathrm{sin}\left(\mathrm{tan}^{−\mathrm{1}} \left(\mathrm{2}{x}\right)\right)\right)−\frac{\mathrm{1}}{\mathrm{2}}{x}^{\mathrm{3}} \mathrm{cos}\left(\mathrm{tan}^{−\mathrm{1}} \left(\mathrm{2}{x}\right)\right)\right) \\ $$ Terms of…
Question Number 213423 by efronzo1 last updated on 05/Nov/24 $$\:\:\lfloor\:\frac{\mathrm{1}}{\mathrm{2}}\mathrm{x}−\mathrm{1}\rfloor\:+\:\lfloor\:\frac{\mathrm{2}}{\mathrm{2}}\mathrm{x}−\mathrm{2}\rfloor+\lfloor\frac{\mathrm{3}}{\mathrm{2}}\mathrm{x}−\mathrm{3}\rfloor+…+\lfloor\frac{\mathrm{100}}{\mathrm{2}}\mathrm{x}−\mathrm{100}\rfloor\:\leqslant\mathrm{10100} \\ $$$$\:\:\mathrm{for}\:\mathrm{x}\:\mathrm{non}\:\mathrm{negative}\:\mathrm{integers}. \\ $$$$\:\:\:\mathrm{find}\:\mathrm{the}\:\mathrm{possible}\:\mathrm{value}\:\mathrm{of}\:\mathrm{x} \\ $$ Answered by golsendro last updated on 05/Nov/24 $$\:\:\:\underline{ }…
Question Number 213342 by Spillover last updated on 03/Nov/24 Commented by MrGaster last updated on 03/Nov/24 $$\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\begin{array}{|c|}{\:\:\int{H}_{{x}} ^{\sqrt{\pi}} {dx}=\frac{{H}_{{x}} ^{\sqrt{\pi}+\mathrm{1}} }{\:\sqrt{\pi}+\mathrm{1}}+{C}\:\:}\\\hline\end{array} \\ $$ Terms of…
Question Number 213343 by Spillover last updated on 03/Nov/24 Answered by MrGaster last updated on 03/Nov/24 $$=\int_{\mathrm{0}} ^{\mathrm{1}} \int_{\mathrm{0}} ^{\mathrm{1}} \mathrm{sin}\left({x}\right)\mathrm{sin}\left({y}\right)\left(\int_{\mathrm{0}} ^{\mathrm{1}} \frac{\mathrm{sin}\left({z}\right)}{{x}+{y}+{z}}{dx}\right){dydx} \\ $$$$=\int_{\mathrm{0}}…
Question Number 213376 by RoseAli last updated on 03/Nov/24 $$\int\frac{{dx}}{\:\sqrt{\left(\mathrm{4}{x}^{\mathrm{2}} +\mathrm{1}\right)^{\mathrm{3}} }} \\ $$ Commented by Frix last updated on 03/Nov/24 $$\mathrm{Sometimes}\:\mathrm{just}\:\mathrm{use}\:\mathrm{your}\:\mathrm{brain}\:\&\:\mathrm{experience} \\ $$$$\frac{{d}}{{dx}}\left[\frac{{g}\left({x}\right)}{\:\sqrt{\mathrm{4}{x}^{\mathrm{2}} +\mathrm{1}}}\right]=\frac{{g}'\left({x}\right)\left(\mathrm{4}{x}^{\mathrm{2}}…