Question Number 162298 by mathmax by abdo last updated on 28/Dec/21 $$\mathrm{find}\:\int_{\mathrm{0}} ^{\mathrm{1}} \mathrm{lnx}\:\mathrm{ln}\left(\mathrm{1}−\mathrm{x}^{\mathrm{3}} \right)\mathrm{dx} \\ $$ Answered by Ar Brandon last updated on 28/Dec/21…
Question Number 96758 by bobhans last updated on 04/Jun/20 $$\mathrm{Let}\:{x}\in\:\left[\:−\frac{\mathrm{5}\pi}{\mathrm{12}}\:,\:−\frac{\pi}{\mathrm{3}}\:\right]\:.\:\mathrm{The}\:\mathrm{maximum}\: \\ $$$$\mathrm{value}\:\mathrm{of}\:\mathrm{y}\:=\:\mathrm{tan}\:\left({x}+\frac{\mathrm{2}\pi}{\mathrm{3}}\right)−\mathrm{tan}\:\left({x}+\frac{\pi}{\mathrm{6}}\right)\:+\mathrm{cos}\:\left({x}+\frac{\pi}{\mathrm{6}}\right) \\ $$$$\mathrm{is}\:\_\_\_ \\ $$ Commented by john santu last updated on 04/Jun/20 $$\mathrm{set}\:{m}\:=\:−{x}−\frac{\pi}{\mathrm{6}},\:{m}\in\:\left[\:\frac{\pi}{\mathrm{6}},\:\frac{\pi}{\mathrm{4}}\:\right]…
Question Number 96748 by MJS last updated on 04/Jun/20 $$\mathrm{nobody}\:\mathrm{tried}\:\mathrm{question}\:\mathrm{94184}… \\ $$ Commented by bemath last updated on 04/Jun/20 how did mister solve this integral? Terms of Service Privacy Policy…
Question Number 96746 by MJS last updated on 04/Jun/20 $$\int\frac{\sqrt{{x}}}{\left(\mathrm{1}+{x}^{\mathrm{3}} \right)\sqrt{\mathrm{1}−{x}^{\mathrm{3}} }}{dx}=? \\ $$$$\int\frac{\sqrt{{x}}}{\left(\mathrm{1}−{x}^{\mathrm{3}} \right)\sqrt{\mathrm{1}+{x}^{\mathrm{3}} }}{dx}=? \\ $$ Commented by bemath last updated on 04/Jun/20…
Question Number 96705 by bemath last updated on 04/Jun/20 $$\int\:\mathrm{ln}\left(\sqrt{\mathrm{1}−\mathrm{x}}+\sqrt{\mathrm{1}+\mathrm{x}}\right)\:\mathrm{dx}\:=\:? \\ $$ Answered by john santu last updated on 04/Jun/20 $$\mathrm{D}.\mathrm{I}\:\mathrm{method} \\ $$$$\mathrm{I}\:=\:{x}\:\mathrm{ln}\left(\sqrt{\mathrm{1}−{x}}+\sqrt{\mathrm{1}+{x}}\right)\:−\int\:\frac{{x}\left(\frac{−\mathrm{1}}{\mathrm{2}\sqrt{\mathrm{1}−{x}}}+\frac{\mathrm{1}}{\mathrm{2}\sqrt{\mathrm{1}+{x}}}\right)}{\:\sqrt{\mathrm{1}−{x}}+\sqrt{\mathrm{1}+{x}}}\:{dx} \\ $$$$\mathrm{I}\:=\:{x}\:\mathrm{ln}\left(\sqrt{\mathrm{1}−{x}}+\sqrt{\mathrm{1}+{x}}\right)−\frac{\mathrm{1}}{\mathrm{2}}\int\:{x}\left(\frac{\sqrt{\mathrm{1}−{x}^{\mathrm{2}}…
Question Number 162243 by cortano last updated on 28/Dec/21 $$\:\:\:\int_{\:\mathrm{0}} ^{\:\mathrm{1}} \:\left(\frac{\left(\mathrm{ln}\:{x}\right)^{\mathrm{4}} }{\:\sqrt{{x}}\:}\right)\:{dx}\:=? \\ $$ Answered by aleks041103 last updated on 28/Dec/21 $$\frac{{dx}}{\:\sqrt{{x}}}=\mathrm{2}{d}\left(\sqrt{{x}}\right) \\ $$$$\Rightarrow{I}=\int_{\:\mathrm{0}}…
Question Number 96699 by Rio Michael last updated on 04/Jun/20 $$\int\:\frac{\mathrm{tan}^{\mathrm{3}} \left(\mathrm{ln}\:{x}\right)}{{x}}\:{dx}\:=\:?? \\ $$ Commented by bobhans last updated on 04/Jun/20 $$\mathrm{u}=\mathrm{ln}\left(\mathrm{x}\right)\:\Rightarrow\:\int\:\mathrm{tan}\:^{\mathrm{3}} \mathrm{u}\:\mathrm{du}\:=\:\int\left(\mathrm{sec}\:^{\mathrm{2}} \mathrm{u}−\mathrm{1}\right)\mathrm{tan}\:\mathrm{u}\:\mathrm{du} \\…
Question Number 162238 by amin96 last updated on 27/Dec/21 $$\int_{\mathrm{0}} ^{\mathrm{1}} \frac{\mathrm{1}}{\boldsymbol{\mathrm{x}}^{\mathrm{7}} +\mathrm{1}}\boldsymbol{\mathrm{dx}}=? \\ $$ Answered by Ar Brandon last updated on 27/Dec/21 $${z}^{\mathrm{7}} +\mathrm{1}=\mathrm{0}\Rightarrow{z}_{{k}}…
Question Number 96693 by 175 last updated on 03/Jun/20 Answered by abdomathmax last updated on 04/Jun/20 $$\mathrm{A}_{\mathrm{n}} =\int\:\:\frac{\mathrm{dx}}{\mathrm{1}+\mathrm{tan}^{\mathrm{n}} \mathrm{x}}\:=_{\mathrm{tanx}\:=\mathrm{t}} \:\:\:\:\int\:\:\frac{\mathrm{dt}}{\left(\mathrm{1}+\mathrm{t}^{\mathrm{2}} \right)\left(\mathrm{1}+\mathrm{t}^{\mathrm{n}} \right)} \\ $$$$\mathrm{let}\:\mathrm{decompose}\:\mathrm{F}\left(\mathrm{t}\right)\:=\frac{\mathrm{1}}{\left(\mathrm{t}^{\mathrm{2}} +\mathrm{1}\right)\left(\mathrm{t}^{\mathrm{n}}…
Question Number 31145 by Joel578 last updated on 03/Mar/18 $$\mathrm{Given} \\ $$$$\int_{\mathrm{0}} ^{\mathrm{1}} \:{f}\left({x}\right)\:{dx}\:=\:\begin{pmatrix}{\mathrm{2018}}\\{\:\:\:\:\mathrm{0}}\end{pmatrix}\:+\:\frac{\mathrm{1}}{\mathrm{2}}\begin{pmatrix}{\mathrm{2018}}\\{\:\:\:\:\mathrm{1}}\end{pmatrix}\:+\:\frac{\mathrm{1}}{\mathrm{3}}\begin{pmatrix}{\mathrm{2018}}\\{\:\:\:\:\mathrm{2}}\end{pmatrix}\:+\:…\:+\:\frac{\mathrm{1}}{\mathrm{2019}}\begin{pmatrix}{\mathrm{2018}}\\{\mathrm{2018}}\end{pmatrix} \\ $$$$\int_{\mathrm{0}} ^{\mathrm{1}} \:{g}\left({x}\right)\:{dx}\:=\:\begin{pmatrix}{\mathrm{2018}}\\{\:\:\:\:\mathrm{0}}\end{pmatrix}\:−\:\frac{\mathrm{1}}{\mathrm{2}}\begin{pmatrix}{\mathrm{2018}}\\{\:\:\:\:\mathrm{1}}\end{pmatrix}\:+\:\frac{\mathrm{1}}{\mathrm{3}}\begin{pmatrix}{\mathrm{2018}}\\{\:\:\:\:\mathrm{2}}\end{pmatrix}\:−\:…\:+\:\frac{\mathrm{1}}{\mathrm{2019}}\begin{pmatrix}{\mathrm{2018}}\\{\mathrm{2018}}\end{pmatrix} \\ $$$${h}\left({x}\right)\:\mathrm{is}\:\mathrm{an}\:\mathrm{odd}\:\mathrm{function} \\ $$$$\mathrm{Then}\:\mathrm{what}\:\mathrm{is}\:\mathrm{the}\:\mathrm{value}\:\mathrm{of}\:\int_{−\mathrm{3}} ^{\:\mathrm{3}} \:{f}\left({x}\right).{g}\left({x}\right).{h}\left({x}\right)\:{dx}\:? \\…