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Category: Integration

1-x-dx-

Question Number 192212 by josemate19 last updated on 11/May/23 $$\int−\mathrm{1}^{{x}} {dx} \\ $$ Answered by Frix last updated on 12/May/23 $$\int−\mathrm{1}^{{x}} {dx}=−\int\mathrm{1}^{{x}} {dx}=−\int{dx}=−{x}+{C} \\ $$$$\int\left(−\mathrm{1}\right)^{{x}}…

dx-1-x-2-2x-2-1-1-4-

Question Number 126604 by bramlexs22 last updated on 22/Dec/20 $$\:\int\:\frac{{dx}}{\left(\mathrm{1}−{x}^{\mathrm{2}} \right)\:\sqrt[{\mathrm{4}}]{\mathrm{2}{x}^{\mathrm{2}} −\mathrm{1}}}\:? \\ $$ Answered by liberty last updated on 22/Dec/20 $${Y}=\int\:\frac{\left(\mathrm{1}−{x}^{\mathrm{2}} \right)}{\left(\mathrm{1}−{x}^{\mathrm{2}} \right)^{\mathrm{2}} \:\sqrt[{\mathrm{4}}]{\mathrm{2}{x}^{\mathrm{2}}…

x-sin-x-1-cos-x-dx-

Question Number 61056 by Tawa1 last updated on 28/May/19 $$\int\:\frac{\mathrm{x}\:+\:\mathrm{sin}\left(\mathrm{x}\right)}{\mathrm{1}\:+\:\mathrm{cos}\left(\mathrm{x}\right)}\:\mathrm{dx} \\ $$ Answered by perlman last updated on 28/May/19 $${cos}\left({x}\right)=\mathrm{2}{cos}^{\mathrm{2}} \left(\frac{{x}}{\mathrm{2}}\right)−\mathrm{1} \\ $$$$\frac{{x}+{sin}\left({x}\right)}{\mathrm{1}+{cos}\left({x}\right)}=\frac{{x}+{sin}\left({x}\right)}{\mathrm{2}{cos}^{\mathrm{2}} \left(\frac{{x}}{\mathrm{2}}\right)}=\frac{{x}}{\mathrm{2}{cos}^{\mathrm{2}} \left(\frac{{x}}{\mathrm{2}}\right)}+\frac{\mathrm{2}{sin}\left(\frac{{x}}{\mathrm{2}}\right){cos}\left(\frac{{x}}{\mathrm{2}}\right)}{\mathrm{2}{cos}^{\mathrm{2}}…

sin-x-cos-x-sin-x-dx-

Question Number 126586 by benjo_mathlover last updated on 22/Dec/20 $$\:\:\int\:\frac{\mathrm{sin}\:{x}}{\mathrm{cos}\:{x}−\mathrm{sin}\:{x}}\:{dx}\:? \\ $$ Answered by liberty last updated on 22/Dec/20 $${partial}\:{fraction} \\ $$$$\:\frac{\mathrm{sin}\:{x}}{\mathrm{cos}\:{x}−\mathrm{sin}\:{x}}\:=\:{P}\left(\frac{\mathrm{cos}\:{x}−\mathrm{sin}\:{x}}{\mathrm{cos}\:{x}−\mathrm{sin}\:{x}}\right)+{Q}\:\frac{\frac{{d}}{{dx}}\left(\mathrm{cos}\:{x}−\mathrm{sin}\:{x}\right)}{\mathrm{cos}\:{x}−\mathrm{sin}\:{x}} \\ $$$$\Leftrightarrow\:\mathrm{sin}\:{x}\:=\:{P}\left(\mathrm{cos}\:{x}−\mathrm{sin}\:{x}\right)+{Q}\left(−\mathrm{sin}\:{x}−\mathrm{cos}\:{x}\right) \\…