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} \\ $$$${Let}\:\:\:\mathrm{2}{x}=\mathrm{sinh}\:\theta\:\:\:\:\:\:\:\:\:\:\:{dx}=\:\frac{\mathrm{cosh}\:\theta{d}\theta}{\mathrm{2}} \\ $$$$\int_{\mathrm{0}} ^{\sqrt{\mathrm{2}}}…
Question Number 61388 by maxmathsup by imad last updated on 02/Jun/19 $${calculate}\:\int_{\mathrm{0}} ^{\mathrm{1}} \:\:\frac{{sin}\left({lnx}\right)}{{lnx}}\:{dx}\:. \\ $$ Commented by perlman last updated on 02/Jun/19 $${let}\:{u}={ln}\left({x}\right) \\…
Question Number 61386 by maxmathsup by imad last updated on 02/Jun/19 $${find}\:{the}\:{value}\:{of}\:\int_{−\infty} ^{+\infty} \:\:\frac{{ln}\left(\mathrm{1}+{x}^{\mathrm{2}} \right)}{\mathrm{1}+{x}^{\mathrm{2}} }\:{dx} \\ $$ Commented by perlman last updated on 02/Jun/19…
Question Number 192453 by Spillover last updated on 18/May/23 $${Find}\:\:\:\:\:\underset{{n}\rightarrow\infty} {\mathrm{lim}}\:\left(\frac{\mathrm{1}}{{n}^{\mathrm{2}} }+\frac{\mathrm{2}}{{n}^{\mathrm{2}} }+\frac{\mathrm{3}}{{n}^{\mathrm{2}} }+…\frac{{n}}{{n}^{\mathrm{2}} }\right) \\ $$ Answered by senestro last updated on 18/May/23 $$\mathrm{1}/\mathrm{2}…
Question Number 192452 by Spillover last updated on 18/May/23 $${Evaluate}\:\:\:\:\:\int_{\mathrm{0}} ^{\mathrm{1}} \mathrm{2}^{{x}\:} {dx}\:\:\: \\ $$$$ \\ $$ Answered by senestro last updated on 18/May/23 $$\mathrm{1}/\mathrm{ln}\:\mathrm{2}…