Question Number 121696 by bemath last updated on 11/Nov/20 $$\:\:\int\:\frac{{x}^{\mathrm{4}} −\mathrm{5}{x}^{\mathrm{3}} +\mathrm{6}{x}^{\mathrm{2}} −\mathrm{18}}{{x}^{\mathrm{3}} −\mathrm{3}{x}^{\mathrm{2}} }\:{dx}\:? \\ $$ Answered by liberty last updated on 11/Nov/20 $$\mathrm{J}=\int\:\left[\left(\mathrm{x}−\mathrm{2}\right)−\frac{\mathrm{18}}{\mathrm{x}^{\mathrm{2}}…
Question Number 121680 by aristarque last updated on 10/Nov/20 $${please}\:{evaluate}\:\int{x}^{{x}} {dx} \\ $$ Answered by Dwaipayan Shikari last updated on 11/Nov/20 $$\int{e}^{{xlogx}} {dx} \\ $$$$=\int\underset{{n}=\mathrm{0}}…
Question Number 121674 by mnjuly1970 last updated on 10/Nov/20 $$\:\:\:\:\:\:\:\:\:…\:\:{advanced}\:\:{calculus}… \\ $$$$\:\:\:\:{prove}\:{that}\:: \\ $$$$ \\ $$$$\:\:\:\:\:\:\:\:\:\:\Omega\:=\int_{\mathrm{0}} ^{\:\infty} \frac{{sin}^{\mathrm{4}} \left({x}\right){ln}\left({x}\right)}{{x}^{\mathrm{2}} }{dx}=\frac{\pi}{\mathrm{4}}\left(\mathrm{1}−\gamma\right) \\ $$$$\gamma:{euler}−{mascheroni}\:{constant} \\ $$$$\:\:\:\:\:\:{m}.{n}.{july}.\mathrm{1970} \\…
Question Number 56107 by Joel578 last updated on 10/Mar/19 $$\int_{−\mathrm{1}} ^{\mathrm{0}} \:\mid{x}\:\mathrm{sin}\:\left(\pi{x}\right)\mid\:{dx} \\ $$ Commented by maxmathsup by imad last updated on 10/Mar/19 $${let}\:{I}\:=\int_{−\mathrm{1}} ^{\mathrm{0}}…
Question Number 187164 by mathlove last updated on 14/Feb/23 Answered by qaz last updated on 14/Feb/23 $$\underset{{p}\rightarrow{ln}\frac{\mathrm{1}}{\mathrm{2}}} {{lim}}\frac{{ln}\mathrm{2}+{ln}\mathrm{2}\:{cos}\:{p}}{\left({cos}\:{ln}\frac{\mathrm{1}}{\:\sqrt{\mathrm{2}}}\right)^{\mathrm{2}} }=\frac{\left(\mathrm{1}+{cos}\:{ln}\mathrm{2}\right){ln}\mathrm{2}}{{cos}\:^{\mathrm{2}} \frac{\mathrm{1}}{\mathrm{2}}{ln}\mathrm{2}}=\frac{\left[\mathrm{2cos}\:^{\mathrm{2}} \frac{\mathrm{1}}{\mathrm{2}}{ln}\mathrm{2}\right]{ln}\mathrm{2}}{{cos}^{\mathrm{2}} \frac{\mathrm{1}}{\mathrm{2}}{ln}\mathrm{2}}=\mathrm{2}{ln}\mathrm{2} \\ $$$$\underset{{p}=\mathrm{0}} {\overset{\infty}…
Question Number 121611 by benjo_mathlover last updated on 10/Nov/20 $$\:\underset{\mathrm{0}} {\overset{\mathrm{2}} {\int}}\:\mathrm{sin}\:\left(\mathrm{2}\pi\mathrm{x}\right)\mathrm{cos}\:\left(\mathrm{5}\pi\mathrm{x}\right)\:\mathrm{dx}\:?\: \\ $$ Commented by mr W last updated on 10/Nov/20 $$\int_{\mathrm{0}} ^{\mathrm{2}} \mathrm{sin}\:\left(\mathrm{2}\pi{x}\right)\mathrm{cos}\:\left(\mathrm{5}\pi{x}\right){dx}…
Question Number 121602 by benjo_mathlover last updated on 09/Nov/20 Answered by liberty last updated on 10/Nov/20 $$\mathrm{I}=\underset{\mathrm{0}} {\overset{\pi/\mathrm{2}} {\int}}\:\frac{\mathrm{dx}}{\mathrm{b}^{\mathrm{2}} \:\mathrm{tan}\:^{\mathrm{2}} \mathrm{x}+\mathrm{1}} \\ $$$$\mathrm{let}\:\phi=\mathrm{b}\:\mathrm{tan}\:\mathrm{x}\:\Rightarrow\mathrm{d}\phi=\mathrm{bsec}\:^{\mathrm{2}} \mathrm{x}\:\mathrm{dx}\: \\…
Question Number 121601 by benjo_mathlover last updated on 09/Nov/20 $$\:\:\int\:\frac{\mathrm{dx}}{\mathrm{x}\:\sqrt{\mathrm{3}+\mathrm{x}^{\mathrm{2}} }}\:? \\ $$ Answered by liberty last updated on 10/Nov/20 $$\mathrm{let}\:\mathrm{x}\:=\:\sqrt{\mathrm{3}}\:\mathrm{tan}\:\gamma\: \\ $$$$\mathrm{I}=\:\int\:\frac{\sqrt{\mathrm{3}}\:\mathrm{sec}\:^{\mathrm{2}} \gamma}{\:\sqrt{\mathrm{3}}\:\mathrm{tan}\:\gamma\:\sqrt{\mathrm{3}+\mathrm{3tan}\:^{\mathrm{2}} \gamma}}\:\mathrm{d}\gamma…
Question Number 187135 by cortano12 last updated on 14/Feb/23 $$\:\underset{\:\:\mathrm{0}} {\overset{\:\:\pi/\mathrm{4}} {\int}}\:\frac{\mathrm{tan}\:^{\mathrm{2}} {x}}{\mathrm{1}+\mathrm{sin}\:{x}}\:{dx}\:=? \\ $$ Answered by horsebrand11 last updated on 14/Feb/23 $$\:{I}=\underset{\:\mathrm{0}} {\overset{\:\pi/\mathrm{4}} {\int}}\:\frac{\mathrm{tan}\:^{\mathrm{2}}…
Question Number 56060 by necx1 last updated on 09/Mar/19 $$\int{e}^{−{x}^{\mathrm{2}} } {dx}\:{as}\:{an}\:{infinite}\:{series}.{Hence} \\ $$$${investigate}\:{its}\:{converge}. \\ $$ Commented by maxmathsup by imad last updated on 09/Mar/19…