Question Number 222858 by MrGaster last updated on 09/Jul/25 $$\mathrm{Prove}: \\ $$$$\int_{\mathrm{0}} ^{\frac{\mathrm{1}}{\mathrm{2}}} \frac{\mathrm{arcsin}^{\mathrm{2}} {x}}{{x}}{dx}=\frac{\pi{i}}{\mathrm{6}}\left(\frac{\pi^{\mathrm{2}} }{\mathrm{36}}−\mathrm{Li}_{\mathrm{2}} \left(\frac{\mathrm{1}+{i}\sqrt{\mathrm{3}}}{\mathrm{2}}\right)\right)−\frac{\mathrm{1}}{\mathrm{3}}\zeta\left(\mathrm{3}\right) \\ $$ Commented by gabthemathguy25 last updated on…
Question Number 222855 by MrGaster last updated on 09/Jul/25 $$\mathrm{Prove}:\int_{−\pi} ^{\pi} {x}\:\mathrm{ln}\:\left(\mathrm{1}+\mathrm{sin}\:{x}\:+\mathrm{cos}\:{x}\right){dx}=\mathrm{2}\pi{G} \\ $$ Answered by MrGaster last updated on 09/Jul/25 Terms of Service Privacy…
Question Number 222848 by MrGaster last updated on 09/Jul/25 $$ \\ $$$$\int_{\mathrm{0}} ^{\mathrm{1}} \frac{\mathrm{arcatn}^{\mathrm{2}} {x}}{{x}}{dx}=\frac{\pi}{\mathrm{2}}{G}−\frac{\mathrm{7}}{\mathrm{8}}\zeta\left(\mathrm{3}\right) \\ $$ Answered by MrGaster last updated on 09/Jul/25 $$\mathrm{cos}\:{x}=\underset{{n}=\mathrm{0}}…
Question Number 222850 by MrGaster last updated on 09/Jul/25 $$\:\:\:\:\:\:\:\int_{\mathrm{0}} ^{\infty} \frac{{x}^{−{x}} {e}^{−{x}} }{\Gamma\left(\mathrm{1}−{x}\right)}{dx} \\ $$ Answered by MrGaster last updated on 09/Jul/25 $$\Gamma\left(\mathrm{1}−{x}\right)\Gamma\left({x}\right)=\frac{\pi}{\mathrm{sin}\left(\pi{x}\right)} \\…
Question Number 222828 by wewji12 last updated on 09/Jul/25 $$\mathrm{vector}\:\mathrm{field}\:\:\overset{\rightarrow} {\boldsymbol{\mathrm{F}}};\mathbb{R}^{\mathrm{3}} \rightarrow\mathbb{R}^{\mathrm{3}} \:,\:{F}_{{h}} \in\mathcal{C}^{\omega} \\ $$$$\mathrm{and}\:\mathrm{Let}'\mathrm{s}\:\mathrm{define}\:\mathrm{as}\:\overset{\rightarrow} {\boldsymbol{\mathrm{A}}}=\overset{\rightarrow} {\bigtriangledown}×\overset{\rightarrow} {\boldsymbol{\mathrm{F}}} \\ $$$$\mathrm{can}\:\mathrm{we}\:\mathrm{find}\:\mathrm{vector}\:\mathrm{field}\:\overset{\rightarrow} {\boldsymbol{\mathrm{F}}}…..??? \\ $$$$\mathrm{Curl}\:\mathrm{and}\:\mathrm{Divergence}\:\:\mathrm{inverse}\:\mathrm{operator}\:\mathrm{dose}\:\mathrm{exist}?? \\…
Question Number 222812 by MrGaster last updated on 08/Jul/25 Answered by MrGaster last updated on 08/Jul/25 $${x}=\mathrm{sinh}\:{u}\Rightarrow{dx}=\mathrm{cosh}\:{udu},\sqrt{\mathrm{1}+{u}^{\mathrm{2}} }=\mathrm{cosh}\:{u} \\ $$$${u}\mid_{{x}=\mathrm{0}} =\mathrm{0},{u}\mid_{{x}=\frac{\mathrm{1}}{\mathrm{2}}} =\mathrm{sinh}^{−\mathrm{1}} \left(\frac{\mathrm{1}}{\mathrm{2}}\right) \\ $$$${u}=\mathrm{sinh}^{−\mathrm{1}}…
Question Number 222829 by hardmath last updated on 08/Jul/25 $$\mathrm{If}\:\:\:\mathrm{f}\left(\mathrm{x}\right)\:=\:\frac{\mathrm{3x}\:+\:\left[\mathrm{x}\right]}{\mathrm{2x}} \\ $$$$\mathrm{Find}\:\:\:\underset{\boldsymbol{\mathrm{x}}\rightarrow−\mathrm{5}^{+} } {\mathrm{lim}}\:\mathrm{f}\left(\mathrm{x}\right)\:−\:\underset{\boldsymbol{\mathrm{x}}\rightarrow−\mathrm{5}^{−} } {\mathrm{lim}}\:\mathrm{f}\left(\mathrm{x}\right)\:=\:? \\ $$ Answered by mehdee7396 last updated on 08/Jul/25…
Question Number 222830 by Tawa11 last updated on 08/Jul/25 Answered by Frix last updated on 08/Jul/25 $$\underset{{i}=\mathrm{1}} {\overset{{k}} {\sum}}…= \\ $$$$=\frac{\mathrm{32}}{\mathrm{9}}\underset{{i}=\mathrm{1}} {\overset{{k}} {\sum}}\frac{{i}^{\mathrm{3}} }{{k}^{\mathrm{4}} }+\mathrm{120}\underset{{i}=\mathrm{1}}…
Question Number 222811 by MrGaster last updated on 08/Jul/25 $$\:\:\:\:\:\:\:\:\:\:\:\:\int_{\mathrm{0}} ^{\frac{\mathrm{1}}{\mathrm{2}}} \frac{\mathrm{ln}\left(\mathrm{2}{x}\right)}{\:\sqrt{\mathrm{1}−{x}^{\mathrm{2}} }}{dx} \\ $$ Answered by MrGaster last updated on 08/Jul/25 Terms of Service…
Question Number 222805 by MrGaster last updated on 08/Jul/25 $$\mathrm{Prove}:\int_{−\infty} ^{\infty} {J}_{\mathrm{0}} \left(\mathrm{2}{x}\right){dx}=\mathrm{1} \\ $$ Answered by MrGaster last updated on 08/Jul/25 $${J}_{\mathrm{0}} \left(\mathrm{2}{x}\right)=\frac{\mathrm{1}}{\mathrm{2}\pi}\int_{−\pi} ^{\pi}…