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Category: Integration

0-pi-2-cos-2x-ln-sin-x-dx-pi-4-solution-1-0-pi-2-2cos-2-x-1-ln-sin-x-dx-2-0-pi-2-cos-2-x-ln-sin-x-dx-0-pi-2-

Question Number 153555 by mnjuly1970 last updated on 08/Sep/21 $$ \\ $$$$\:\:\:\:\:\Omega\::=\:\int_{\mathrm{0}} ^{\:\frac{\pi}{\mathrm{2}}} {cos}\left(\mathrm{2}{x}\right).{ln}\left({sin}\left({x}\right)\right){dx}\overset{?} {=}\:−\frac{\pi}{\mathrm{4}} \\ $$$$\:\:\:\:\:\:\:\:\:\:{solution}\:\left(\mathrm{1}\:\right) \\ $$$$\:\:\:\:\:\Omega\::=\:\int_{\mathrm{0}} ^{\:\frac{\pi}{\mathrm{2}}} \left(\:\mathrm{2}{cos}^{\:\mathrm{2}} \left({x}\right)−\mathrm{1}\right){ln}\left({sin}\left({x}\right)\right){dx} \\ $$$$\:\:\:\:\:\::=\:\mathrm{2}\int_{\mathrm{0}} ^{\:\frac{\pi}{\mathrm{2}}}…

x-2-2x-3-x-2-x-1-dx-

Question Number 88010 by Chi Mes Try last updated on 07/Apr/20 $$\int\frac{{x}^{\mathrm{2}} +\mathrm{2}{x}+\mathrm{3}}{\:\sqrt{{x}^{\mathrm{2}} +{x}+\mathrm{1}}}{dx} \\ $$ Commented by niroj last updated on 07/Apr/20 $$\:\:\:\int\frac{\mathrm{x}^{\mathrm{2}} +\mathrm{2x}+\mathrm{3}}{\:\sqrt{\mathrm{x}^{\mathrm{2}}…

0-1-4-x-3-1-dx-

Question Number 88007 by Mikael_786 last updated on 07/Apr/20 $$\underset{\mathrm{0}} {\overset{\mathrm{1}} {\int}}\sqrt{\sqrt{\frac{\mathrm{4}}{{x}}−\mathrm{3}}−\mathrm{1}}{dx}=? \\ $$ Answered by MJS last updated on 07/Apr/20 $${t}=\sqrt{\sqrt{\frac{\mathrm{4}}{{x}}−\mathrm{3}}−\mathrm{1}}\:\rightarrow\:{dx}=−{x}^{\mathrm{2}} \sqrt{\frac{\mathrm{4}}{{x}}−\mathrm{3}}\sqrt{\sqrt{\frac{\mathrm{4}}{{x}}−\mathrm{3}}−\mathrm{1}}{dt} \\ $$$$\Rightarrow…

Determine-all-functions-f-0-1-such-that-x-0-1-f-x-f-x-f-0-f-1-

Question Number 88003 by Ar Brandon last updated on 07/Apr/20 $${Determine}\:{all}\:{functions}\:{f}\left[\mathrm{0},\mathrm{1}\right]\rightarrow\Omega \\ $$$${such}\:{that}\:\forall{x}\in\left[\mathrm{0},\mathrm{1}\right]\:{f}\:'\left({x}\right)+{f}\left({x}\right)={f}\left(\mathrm{0}\right)+{f}\left(\mathrm{1}\right) \\ $$ Commented by mr W last updated on 07/Apr/20 $${f}\left(\mathrm{0}\right)+{f}\left(\mathrm{1}\right)={k} \\…