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 $$\mathrm{1}/\mathrm{ln}\:\mathrm{2}…
Question Number 192454 by Spillover last updated on 18/May/23 $${Evaluate}\:{the}\:{following}\:{improper}\:{intergrals} \\ $$$$\int_{\frac{\pi}{\mathrm{4}}} ^{\frac{\pi}{\mathrm{2}}} \mathrm{sec}\:{xdx}\:\:{if}\:{it}\:{convergent} \\ $$ Terms of Service Privacy Policy Contact: info@tinkutara.com
Question Number 61349 by tanmay last updated on 01/Jun/19 Commented by MJS last updated on 01/Jun/19 $$…\mathrm{harder}\:\mathrm{than}\:\mathrm{I}\:\mathrm{thought} \\ $$$$\int\frac{\sqrt{{t}^{\mathrm{2}} −\mathrm{5}}}{{t}−{a}}{dt}= \\ $$$$\:\:\:\:\left[{t}=\frac{\sqrt{\mathrm{5}}}{\mathrm{sin}\:{u}}\:\Leftrightarrow\:{u}=\mathrm{arcsin}\:\frac{\sqrt{\mathrm{5}}}{{t}}\:\rightarrow\:{dt}=−\frac{{t}\sqrt{{t}^{\mathrm{2}} −\mathrm{5}}}{\:\sqrt{\mathrm{5}}}{du}\right] \\ $$$$\:\:\:\:\:\mathrm{now}\:\mathrm{the}\:\mathrm{root}\:\mathrm{is}\:\mathrm{gone},\:\mathrm{but}\:\mathrm{we}\:\mathrm{must}\:\mathrm{make}…
Question Number 126879 by bramlexs22 last updated on 25/Dec/20 $$\:\int\:\frac{{x}^{\mathrm{4}} }{\mathrm{1}+{x}^{\mathrm{8}} }\:{dx}\:? \\ $$ Answered by Lordose last updated on 25/Dec/20 $$\Omega\:=\:\int^{\:} \frac{\mathrm{x}^{\mathrm{4}} }{\mathrm{1}+\mathrm{x}^{\mathrm{8}} }\mathrm{dx}\:=\:\underset{\mathrm{n}=\mathrm{0}}…