Question Number 126003 by mnjuly1970 last updated on 16/Dec/20 $$\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:…{nice}\:\:{integral}… \\ $$$$\:\:\:{prove}\:\:{that}\::: \\ $$$$\:\:\:\:\:\phi=\int_{\mathrm{0}} ^{\:\frac{\pi}{\mathrm{2}}} {tan}\left({x}\right){ln}\left({sin}\left({x}\right)\right){ln}\left({cos}\left({x}\right)\right){dx}=\frac{\zeta\left(\mathrm{3}\right.}{\mathrm{8}} \\ $$$$\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\mathscr{G}{ood}\:{luck} \\ $$ Answered by Olaf last updated…
Question Number 126000 by bramlexs22 last updated on 16/Dec/20 $$\:\:\:\int\:\frac{{dx}}{\left({x}−\mathrm{1}\right)\sqrt{{x}^{\mathrm{2}} −\mathrm{2}{x}}}\:? \\ $$ Answered by Dwaipayan Shikari last updated on 16/Dec/20 $$\int\frac{\mathrm{1}}{\left({x}−\mathrm{1}\right)\sqrt{\left({x}−\mathrm{1}\right)^{\mathrm{2}} −\mathrm{1}}}{dx}\:=\:{sec}^{−\mathrm{1}} \left({x}−\mathrm{1}\right)+{C} \\…
Question Number 125999 by bramlexs22 last updated on 16/Dec/20 Commented by bramlexs22 last updated on 16/Dec/20 $${thanks}\:{both}\:{sir} \\ $$ Answered by mr W last updated…
Question Number 125996 by bramlexs22 last updated on 16/Dec/20 $$\:{If}\:{z}^{\mathrm{3}} ={x}^{\mathrm{2}} −{y}^{\mathrm{2}} \:,\:\rightarrow\begin{cases}{\frac{{dx}}{{dt}}=\mathrm{3}}\\{\frac{{dy}}{{dt}}=\mathrm{2}}\end{cases} \\ $$$${find}\:\frac{{dz}}{{dt}}\:{when}\:{x}=\mathrm{4}\:{and}\:{y}=\mathrm{1} \\ $$ Answered by Olaf last updated on 16/Dec/20 $${z}^{\mathrm{3}}…
Question Number 125997 by bramlexs22 last updated on 16/Dec/20 $${Find}\:{the}\:{Riemann}\:{sum}\:{for}\:{the} \\ $$$${given}\:{function}\:{with}\:{the}\:{specified} \\ $$$${number}\:{of}\:{intervals}\:{using}\:{left} \\ $$$${endpoints}\:{f}\left({x}\right)=\:\mathrm{4ln}\:{x}+\mathrm{2}{x}\:;\:\mathrm{1}\leqslant{x}\leqslant\mathrm{4} \\ $$$${n}=\mathrm{7}\:.\:{Round}\:{your}\:{answer}\:{to}\:{two} \\ $$$${decimal}\:{places}\:? \\ $$ Answered by liberty…
Question Number 191529 by MATHEMATICSAM last updated on 25/Apr/23 $$\mathrm{If}\:\frac{{x}\:−\:{y}}{{x}\sqrt{{y}}\:+\:{y}\sqrt{{x}}}\:=\:\frac{\mathrm{1}}{\:\sqrt{{x}}}\:;\:\left({x}\:>\:\mathrm{0}\:\mathrm{and}\:{y}\:>\:\mathrm{0}\right)\:\mathrm{then} \\ $$$$\mathrm{Find}\:\mathrm{the}\:\mathrm{value}\:\mathrm{of}\:\frac{{x}}{{y}}\:. \\ $$ Answered by mehdee42 last updated on 25/Apr/23 $$\frac{\left(\sqrt{{x}}−\sqrt{{y}}\right)\left(\sqrt{{x}}+\sqrt{{y}}\right)}{\:\sqrt{{xy}}\left(\sqrt{{x}}+\sqrt{{y}}\right)}=\frac{\mathrm{1}}{\:\sqrt{{x}}} \\ $$$${x}−\sqrt{{xy}}=\sqrt{{xy}}\Rightarrow{x}=\mathrm{2}\sqrt{{xy}}\Rightarrow\frac{{x}}{{y}}=\mathrm{4}\:\checkmark \\…
Question Number 125994 by Dwaipayan Shikari last updated on 16/Dec/20 $$\frac{\underset{{n}=\mathrm{0}} {\overset{\infty} {\sum}}{e}^{−{n}^{\mathrm{2}} } }{\underset{{n}=\mathrm{0}} {\overset{\infty} {\sum}}{e}^{−\mathrm{2}{n}^{\mathrm{2}} } } \\ $$ Answered by Olaf last…
Question Number 60459 by tanmay last updated on 21/May/19 Commented by bhanukumarb2@gmail.com last updated on 21/May/19 $${source}\:{of}\:{question} \\ $$ Commented by tanmay last updated on…
Question Number 191528 by mnjuly1970 last updated on 25/Apr/23 $$ \\ $$$$\:\:\:\:\:\:{find}\:\:{the}\:\:{value}\:\:{of}\:\:{the} \\ $$$$\:\:\:\:\:\:\:{following}\:\:{series}\:. \\ $$$$ \\ $$$$\:\:\:\:\:\:\:\:\Omega=\:\underset{{n}=\mathrm{1}} {\overset{\infty} {\sum}}\frac{\:{cos}\left(\frac{{n}\pi}{\mathrm{4}}\:\right)}{{n}^{\:\mathrm{2}} }\:=? \\ $$ Answered by…
Question Number 125995 by liberty last updated on 16/Dec/20 Commented by benjo_mathlover last updated on 16/Dec/20 $${f}\left({x}\right)=\:\begin{cases}{{x}\:;\:\mathrm{0}\leqslant{x}\leqslant\mathrm{1}}\\{−{x}\:;\:−\mathrm{1}\leqslant{x}\leqslant\mathrm{0}}\end{cases} \\ $$$$\:{f}\:'\left(−\mathrm{1}\right)\:=\:\underset{{h}\rightarrow\mathrm{0}} {\mathrm{lim}}\:\frac{{f}\left(−\mathrm{1}+{h}\right)−{f}\left(−\mathrm{1}\right)}{{h}} \\ $$$${f}\:'\left(−\mathrm{1}\right)=\:\underset{{h}\rightarrow\mathrm{0}} {\mathrm{lim}}\frac{−\left(−\mathrm{1}+{h}\right)−\left(\mathrm{1}\right)}{{h}} \\ $$$$\:{f}\:'\left(−\mathrm{1}\right)=\:\underset{{h}\rightarrow\mathrm{0}}…