Question Number 62908 by aliesam last updated on 26/Jun/19 $$\int\frac{{arctan}\left({x}\right)}{{x}}{dx} \\ $$ Commented by mathmax by abdo last updated on 26/Jun/19 $$\left.\:\left.{for}\:{all}\:{x}\:{from}\:{R}\:\:\:\:{u}\rightarrow\frac{{arctanu}}{{u}}{is}\:{integrable}\:{on}\:\right]\mathrm{0},{x}\right]{let}\:{f}\left({t}\right)\:=\int_{\mathrm{0}} ^{{x}} \:\frac{{arctan}\left({tu}\right)}{{u}}{du} \\…
Question Number 62906 by Prithwish sen last updated on 26/Jun/19 $$\int\mathrm{x}^{\mathrm{x}} \mathrm{dx} \\ $$ Commented by mathmax by abdo last updated on 27/Jun/19 $${let}\:{A}\:=\int\:{x}^{{x}} \:{dx}\:\Rightarrow{A}\:=\int\:{e}^{{xln}\left({x}\right)}…
Question Number 128417 by liberty last updated on 07/Jan/21 $$\:\:\:\:\:\:\:\:\:\eta\:=\:\int_{\mathrm{0}} ^{\:\mathrm{1}} \:\mathrm{x}^{\mathrm{3}} \:\left(\mathrm{1}−\mathrm{x}^{\mathrm{3}} \right)^{\mathrm{n}−\mathrm{1}} \:\mathrm{dx}\: \\ $$ Answered by Dwaipayan Shikari last updated on 07/Jan/21…
Question Number 128408 by bramlexs22 last updated on 07/Jan/21 $$\rho\:=\:\int\:\frac{\mathrm{sin}\:\left(\mathrm{4}{x}\right)}{\mathrm{sin}\:^{\mathrm{4}} \left({x}\right)+\mathrm{cos}\:^{\mathrm{4}} \left({x}\right)}\:{dx}\: \\ $$ Answered by mr W last updated on 07/Jan/21 $$=\int\frac{\mathrm{sin}\:\left(\mathrm{4}{x}\right)}{\mathrm{1}−\mathrm{2}\:\mathrm{sin}^{\mathrm{2}} \:{x}\:\mathrm{cos}^{\mathrm{2}} \:{x}}{dx}…
Question Number 62856 by mathmax by abdo last updated on 26/Jun/19 $${let}\:{f}\left(\lambda\right)\:=\int_{\mathrm{0}} ^{+\infty} \:\:\:\frac{{x}^{\mathrm{4}} }{{x}^{\mathrm{6}} \:+\lambda^{\mathrm{6}} }\:{dx}\:\:\:{with}\:\lambda>\mathrm{0} \\ $$$$\left.\mathrm{1}\right)\:{calculate}\:\:{f}\left(\lambda\right) \\ $$$$\left.\mathrm{2}\right)\:{calculate}\:{also}\:{g}\left(\lambda\right)\:=\int_{\mathrm{0}} ^{\infty} \:\:\frac{{x}^{\mathrm{4}} }{\left({x}^{\mathrm{6}} \:+\lambda^{\mathrm{6}}…
Question Number 62855 by mathmax by abdo last updated on 26/Jun/19 $${find}\:\int\:\:\left(\frac{{x}^{\mathrm{4}} }{\mathrm{1}+{x}^{\mathrm{6}} }\right)^{\mathrm{2}} \:{dx} \\ $$$$\left.\mathrm{2}\right)\:{calculate}\:\int_{\mathrm{0}} ^{\mathrm{1}} \:\:\:\frac{{x}^{\mathrm{8}} }{\left(\mathrm{1}+{x}^{\mathrm{6}} \right)^{\mathrm{2}} }{dx} \\ $$$$\left.\mathrm{3}\right)\:{calculate}\:\int_{\mathrm{0}} ^{+\infty}…
Question Number 128385 by n0y0n last updated on 06/Jan/21 $$\int_{\mathrm{1}} ^{\:\pi} \begin{vmatrix}{\mathrm{x}^{\mathrm{3}} }&{\mathrm{lnx}}&{\mathrm{sinx}}\\{\mathrm{3x}^{\mathrm{2}} }&{\frac{\mathrm{1}}{\mathrm{x}}}&{\mathrm{cosx}}\\{\mathrm{6}}&{\mathrm{2x}^{−\mathrm{3}} }&{−\mathrm{cosx}}\end{vmatrix}\mathrm{dx}\:=?\: \\ $$ Answered by mathmax by abdo last updated on…
Question Number 128373 by BHOOPENDRA last updated on 06/Jan/21 $$\int_{\mathrm{0}} ^{\mathrm{1}} \:{x}^{\frac{\mathrm{3}}{\mathrm{2}}} \left(\mathrm{1}−{x}\right)^{\frac{\mathrm{1}}{\mathrm{2}}} \:{dx} \\ $$ Answered by Dwaipayan Shikari last updated on 06/Jan/21 $$\int_{\mathrm{0}}…
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Question Number 62833 by aliesam last updated on 25/Jun/19 $$\int\frac{{cos}\left({x}\right)}{{x}}\:{dx} \\ $$$$ \\ $$$$\int\sqrt{{sin}\left({x}\right)\:}\:{dx} \\ $$$$ \\ $$$$\int\sqrt{\mathrm{1}−{k}^{\mathrm{2}} {sin}^{\mathrm{2}} \left({x}\right)}\:{dx}\:\:\:\:\:\:{k}:{constant} \\ $$ Commented by mathmax…