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Category: Integration

cosx-senx-dx-

Question Number 141367 by cesarL last updated on 18/May/21 $$\int\left(\sqrt{{cosx}\centerdot{senx}}\right){dx} \\ $$ Answered by MJS_new last updated on 18/May/21 $$\mathrm{if}\:\mathrm{sen}\:{x}\:=\mathrm{sin}\:{x} \\ $$$$\int\sqrt{\mathrm{cos}\:{x}\:\mathrm{sin}\:{x}}\:{dx}=\frac{\sqrt{\mathrm{2}}}{\mathrm{2}}\int\sqrt{\mathrm{sin}\:\mathrm{2}{x}}\:{dx}= \\ $$$$\:\:\:\:\:\left[{t}={x}−\frac{\pi}{\mathrm{4}}\:\rightarrow\:{dx}={dt}\right] \\…

sen-2-cos-4-2-d-

Question Number 141362 by cesarL last updated on 17/May/21 $$\int{sen}\left(\mathrm{2}\theta\right){cos}^{\mathrm{4}} \left(\mathrm{2}\theta\right){d}\theta \\ $$ Answered by bemath last updated on 17/May/21 $$\int\:\mathrm{cos}\:^{\mathrm{4}} \left(\mathrm{2}\theta\right)\:\mathrm{sin}\:\left(\mathrm{2}\theta\right)\:{d}\theta= \\ $$$$−\frac{\mathrm{1}}{\mathrm{2}}\int\:\mathrm{cos}\:^{\mathrm{4}} \left(\mathrm{2}\theta\right)\:{d}\left(\mathrm{cos}\:\mathrm{2}\theta\right)\:=…

x-2-cos-x-2-dx-

Question Number 141333 by cesarL last updated on 17/May/21 $$\int{x}^{\mathrm{2}} \mathrm{cos}\:\left(\frac{{x}}{\mathrm{2}}\right){dx} \\ $$ Answered by qaz last updated on 17/May/21 $$\int{x}^{\mathrm{2}} \mathrm{cos}\:\frac{{x}}{\mathrm{2}}{dx} \\ $$$$=\mathrm{2}{x}^{\mathrm{2}} \mathrm{sin}\:\frac{{x}}{\mathrm{2}}−\mathrm{4}\int{x}\mathrm{sin}\:\frac{{x}}{\mathrm{2}}{dx}…