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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} \\ $$$$\frac{{d}}{{dx}}\left[\frac{{g}\left({x}\right)}{\:\sqrt{\mathrm{4}{x}^{\mathrm{2}} +\mathrm{1}}}\right]=\frac{{g}'\left({x}\right)\left(\mathrm{4}{x}^{\mathrm{2}}…
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Question Number 213216 by Spillover last updated on 01/Nov/24 Answered by MrGaster last updated on 01/Nov/24 $$=\int_{\mathrm{0}} ^{\infty} \frac{{x}^{\mathrm{3}} }{\left({x}+\mathrm{1}\right)^{\mathrm{2}} \left({x}^{\mathrm{2}} +\mathrm{1}\right)\left({x}^{\mathrm{3}} +\mathrm{1}\right)}{dx}+\int_{\mathrm{0}} ^{\infty} \frac{\mathrm{log}^{\mathrm{2}}…
Question Number 213217 by Spillover last updated on 01/Nov/24 Answered by MrGaster last updated on 01/Nov/24 $$\underset{{n}=\mathrm{1}} {\overset{\infty} {\sum}}\frac{\begin{pmatrix}{\mathrm{2}{n}}\\{{n}}\end{pmatrix}+{n}}{\mathrm{4}^{{n}} \left(\mathrm{2}{n}+\mathrm{1}\right)^{\mathrm{2}} }=\underset{{n}=\mathrm{1}} {\overset{\infty} {\sum}}\frac{\begin{pmatrix}{\mathrm{2}{n}}\\{{n}}\end{pmatrix}}{\mathrm{4}^{{n}} \left(\mathrm{2}{n}+\mathrm{1}\right)^{\mathrm{2}} }+\underset{{n}=\mathrm{1}}…
Question Number 213138 by Spillover last updated on 31/Oct/24 Commented by Frix last updated on 31/Oct/24 $$\mathrm{We}\:\mathrm{must}\:\mathrm{first}\:\mathrm{solve}\:\int\mathrm{sin}\:{t}^{\mathrm{3}} \:{dt}\:\mathrm{which}\:\mathrm{might} \\ $$$$\mathrm{be}\:\mathrm{possible}\:\mathrm{using} \\ $$$$\mathrm{sin}\:{t}^{\mathrm{3}} \:=\frac{\mathrm{e}^{\mathrm{i}{t}^{\mathrm{3}} } −\mathrm{e}^{−\mathrm{i}{t}^{\mathrm{3}}…
Question Number 213139 by Spillover last updated on 31/Oct/24 Answered by MrGaster last updated on 31/Oct/24 $${let}\:{u}=\sqrt{{x}}+\sqrt{{y}}+{z}\Rightarrow{dy}=\frac{\mathrm{1}}{\mathrm{2}\sqrt{{x}}}{dx}+\frac{\mathrm{1}}{\mathrm{2}\sqrt{{y}}}{dy}+\frac{\mathrm{1}}{\mathrm{2}\sqrt{{z}}}{dx} \\ $$$$\Rightarrow\mathrm{0}\leq{u}\leq\frac{\mathrm{3}\sqrt{\pi}}{\mathrm{4}} \\ $$$$\Rightarrow{dxdydz}=\mathrm{8}{u}^{\mathrm{2}} {du} \\ $$$$\Leftrightarrow\int_{\mathrm{0}} ^{\frac{\mathrm{3}\sqrt{\pi}}{\mathrm{4}}}…