Question Number 132162 by Dwaipayan Shikari last updated on 11/Feb/21 $$\underset{{n}=\mathrm{1}} {\overset{\infty} {\sum}}\frac{{sin}\left({n}\right)}{{n}^{\mathrm{2}} } \\ $$ Answered by mnjuly1970 last updated on 11/Feb/21 $$\frac{\mathrm{1}}{\mathrm{2}{i}}\left[{li}_{\mathrm{2}} \left({e}^{{i}}…
Question Number 1088 by 112358 last updated on 10/Jun/15 $${Solve}\:{the}\:{following}\:{integral} \\ $$$${equation}\:{for}\:{f}\left({x}\right): \\ $$$$\int_{\mathrm{0}} ^{\:{x}} {f}\left({t}\right){dt}=\mathrm{3}{f}\left({x}\right)+{k} \\ $$$${where}\:{k}\:{is}\:{a}\:{constant}.\: \\ $$ Answered by prakash jain last…
Question Number 66620 by Mohamed Amine Bouguezzoul last updated on 18/Aug/19 $${find}\:\underset{{n}\rightarrow\infty} {\mathrm{lim}}\:{I}_{{n}} \\ $$$${I}_{{n}} =\int_{\mathrm{0}} ^{\infty} \frac{{dx}}{\left(\mathrm{1}+\mathrm{coth}\:\left({nx}\right)\right)^{{n}} }\:,{n}\geqslant\mathrm{1} \\ $$$$ \\ $$ Commented by…
Question Number 1085 by 123456 last updated on 10/Jun/15 $$\mathrm{I}=\underset{\mathrm{0}} {\overset{\pi} {\int}}\frac{{x}\left(\pi−{x}\right)}{\mathrm{sin}\:{x}}{dx} \\ $$ Commented by 123456 last updated on 10/Jun/15 $${f}\left({x}\right)=\frac{{x}\left(\pi−{x}\right)}{\mathrm{sin}\:{x}} \\ $$$${f}\left(\mathrm{0}^{+} \right)\overset{?}…
Question Number 66621 by lalitchand last updated on 17/Aug/19 Commented by lalitchand last updated on 17/Aug/19 $$\mathrm{8}\:\mathrm{iii} \\ $$ Commented by mr W last updated…
Question Number 132153 by Ñï= last updated on 11/Feb/21 Commented by Ar Brandon last updated on 11/Feb/21 No username ?! ���� Answered by Olaf last updated on 11/Feb/21…
Question Number 1083 by Vishal last updated on 08/Jun/15 $${Let}\:{a},{b},{c},{p}\:{be}\:{rational}\:{numbers}\:{such}\:{that}\:{p}\:{is}\:{not}\:{a}\:{perfect}\:{cube}. \\ $$$${If}\:{a}+{bp}^{\frac{\mathrm{1}}{\mathrm{3}}} +{cp}^{\frac{\mathrm{2}}{\mathrm{3}}} =\mathrm{0},\:{then}\:{prove}\:{that}\:{a}={b}={c}=\mathrm{0}. \\ $$ Answered by prakash jain last updated on 08/Jun/15 $${p}^{\mathrm{1}/\mathrm{3}}…
Question Number 66619 by mr W last updated on 17/Aug/19 $${solve}\:{for}\:{x},{y}\in{R} \\ $$$$\frac{\sqrt{\mathrm{1}+{x}^{\mathrm{2}} }}{\mathrm{ln}\:\left({x}+\sqrt{\mathrm{1}+{x}^{\mathrm{2}} }\right)}=\frac{\sqrt{\mathrm{1}+{y}^{\mathrm{2}} }}{\mathrm{ln}\:\left({y}+\sqrt{\mathrm{1}+{y}^{\mathrm{2}} }\right)} \\ $$ Answered by Smail last updated on…
Question Number 1077 by Vishal last updated on 07/Jun/15 $${Find}\:{the}\:{smallest}\:{number}\:{which}\:{leaves}\:{remainders}\:\mathrm{8}\:{and}\:\mathrm{12}\: \\ $$$${when}\:{divided}\:{by}\:\mathrm{28}\:{and}\:\mathrm{32}\:{respectively}. \\ $$ Commented by prakash jain last updated on 07/Jun/15 $$\mathrm{I}\:\mathrm{assume}\:\mathrm{you}\:\mathrm{mean}\:\mathrm{smallest}\:+\mathrm{ve}\:\mathrm{integer}. \\ $$…
Question Number 132150 by mohammad17 last updated on 11/Feb/21 $${Solve}:\frac{\mid{x}+\mathrm{2}\mid}{{x}−\mathrm{4}}\leqslant\frac{\mathrm{1}}{\mid{x}\mid} \\ $$ Answered by benjo_mathlover last updated on 11/Feb/21 $$\mathrm{case}\left(\mathrm{1}\right)\:\mathrm{x}>\mathrm{0}\:\Rightarrow\:\frac{\mathrm{x}+\mathrm{2}}{\mathrm{x}−\mathrm{4}}\:−\frac{\mathrm{1}}{\mathrm{x}}\leqslant\mathrm{0} \\ $$$$\:\frac{\mathrm{x}^{\mathrm{2}} +\mathrm{2x}−\mathrm{x}+\mathrm{4}}{\mathrm{x}\left(\mathrm{x}−\mathrm{4}\right)}\leqslant\mathrm{0}\:;\:\frac{\mathrm{x}^{\mathrm{2}} +\mathrm{x}+\mathrm{4}}{\mathrm{x}\left(\mathrm{x}−\mathrm{4}\right)}\leqslant\mathrm{0} \\…