Question Number 127032 by mnjuly1970 last updated on 26/Dec/20 Answered by Olaf last updated on 26/Dec/20 $$\left.{i}\right) \\ $$$$\mathrm{1}−\mathrm{2}{r}\mathrm{cos}{x}+{r}^{\mathrm{2}} \:=\:\left({e}^{{ix}} −{r}\right)\left({e}^{−{ix}} −{r}\right) \\ $$$$\mathrm{R}_{{x}} \left({r}\right)\:=\:\frac{\mathrm{1}−{r}^{\mathrm{2}}…
Question Number 127020 by bramlexs22 last updated on 26/Dec/20 $$\:\:{super}\:{nice}\:! \\ $$$$\:\:\:\:\:\:\:\:\:\:\:{show}\:{that}\: \\ $$$$\:\:\:\:\:\:\:\:\:\:\:\zeta\left(\mathrm{6}\right)\:=\:\frac{\pi^{\mathrm{6}} }{\mathrm{945}} \\ $$ Commented by liberty last updated on 26/Dec/20 $${hahaha}\:{very}\:{nice}\:…
Question Number 127017 by mnjuly1970 last updated on 26/Dec/20 $$\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:…{NICE}\:\:\:\:\:{CALCULUS}… \\ $$$$\:\:{prove}\:{that}\::: \\ $$$$\int_{\mathrm{0}} ^{\:\infty} \:\left(\frac{{x}^{\mathrm{2}} {ln}\left(\pi{x}\right)}{\pi^{\pi{x}} }\right){dx} \\ $$$$\:\:=\frac{\mathrm{1}}{\left(\pi{ln}\left(\pi\right)\right)^{\mathrm{3}} }\left[\left(\mathrm{3}−\mathrm{2}\left(\gamma+{ln}\left({ln}\left(\pi\right)\right)\right)\right]\right. \\ $$ Answered by…
Question Number 192543 by peter frank last updated on 20/May/23 Answered by leodera last updated on 20/May/23 $$\Delta\:=\:\int_{\mathrm{0}} ^{\frac{\pi}{\mathrm{2}}} \frac{\mathrm{sin}\:\left({x}\right)}{\mathrm{sin}\:\left({x}\right)+\mathrm{cos}\:\left({x}\right)}{dx} \\ $$$$\Delta\:=\:\frac{\mathrm{1}}{\mathrm{2}}\int_{\mathrm{0}} ^{\frac{\pi}{\mathrm{2}}} \frac{\mathrm{sin}\:\left({x}\right)+\mathrm{cos}\:\left({x}\right)}{\mathrm{sin}\:\left({x}\right)+\mathrm{cos}\:\left({x}\right)}{dx}\:=\:\frac{\mathrm{1}}{\mathrm{2}}\int_{\mathrm{0}} ^{\frac{\pi}{\mathrm{2}}}…
Question Number 61465 by arcana last updated on 02/Jun/19 $$\int_{\mathrm{0}} ^{\mathrm{2}\pi} \frac{\mathrm{1}}{{a}^{\mathrm{2}} {cos}^{\mathrm{2}} \left({t}\right)+{b}^{\mathrm{2}} {sin}^{\mathrm{2}} \left({t}\right)}{dt}=\frac{\mathrm{2}\pi}{{ab}}? \\ $$ Commented by maxmathsup by imad last updated…
Question Number 126997 by bramlexs22 last updated on 26/Dec/20 $$\:\:\underset{\mathrm{0}} {\overset{\mathrm{1}} {\int}}\:\mathrm{arcsin}\:\left(\frac{\mathrm{sin}\:{x}}{\:\sqrt{\mathrm{2}}}\right)\:{dx}\:=? \\ $$ Answered by Evimene last updated on 26/Dec/20 $$\mathrm{solution} \\ $$$$\mathrm{let}\:\sqrt{\mathrm{2}}=\alpha \\…
Question Number 61453 by maxmathsup by imad last updated on 02/Jun/19 $${find}\:\int_{\mathrm{0}} ^{\mathrm{1}} \:\frac{{ln}\left({x}\right){ln}\left(\mathrm{1}+{x}\right)}{{x}}{dx} \\ $$ Answered by Smail last updated on 02/Jun/19 $${A}=\int_{\mathrm{0}} ^{\mathrm{1}}…
Question Number 126986 by mnjuly1970 last updated on 25/Dec/20 $$\:\:\:\:\:\:\:\:\:\:\:…\:{nice}\:\:{calculus}… \\ $$$$\:\:\:\:\:{prove}\:\:{that}\::: \\ $$$$\:\:\:\:\mathrm{I}\::=\:\int_{\mathrm{0}} ^{\:\frac{\pi}{\mathrm{2}}} \frac{\left\{{cot}\left({x}\right)\right\}}{{cot}\left({x}\right)}{dx}=\frac{\mathrm{1}}{\mathrm{2}}\left(\pi−{ln}\left(\frac{{sinh}\left(\pi\right)}{\pi}\right)\right) \\ $$$$\left\{{x}\right\}\:{is}\:{fractional}\:{part}\:{of}\:\:{x}\:.. \\ $$ Answered by Olaf last updated…
Question Number 61408 by aliesam last updated on 02/Jun/19 $$\int_{\mathrm{0}} ^{\pi} \frac{{x}}{{tan}^{\mathrm{2}} \left({x}\right)−\mathrm{1}}\:{dx} \\ $$ Answered by tanmay last updated on 02/Jun/19 $$\int_{\mathrm{0}} ^{\pi} \frac{\pi−{x}}{{tan}^{\mathrm{2}}…
Question Number 192470 by Spillover last updated on 19/May/23 Answered by Spillover last updated on 19/May/23 $$\int_{\mathrm{0}} ^{\sqrt{\mathrm{2}}} \sqrt{\mathrm{1}+\left(\mathrm{2}{x}\right)^{\mathrm{2}} }\:{dx} \\ $$$${Let}\:\:\:\mathrm{2}{x}=\mathrm{sinh}\:\theta\:\:\:\:\:\:\:\:\:\:\:{dx}=\:\frac{\mathrm{cosh}\:\theta{d}\theta}{\mathrm{2}} \\ $$$$\int_{\mathrm{0}} ^{\sqrt{\mathrm{2}}}…