Integration Questions

Question Number 100362 by Dara last updated on 26/Jun/20

$$\int_{\mathrm{0}} ^{\mathrm{1}} \int_{\mathrm{0}} ^{\mathrm{1}} {e}^{\mathrm{2}{x}+{y}} {dydx} \\$$

Answered by smridha last updated on 26/Jun/20

$$\int_{\mathrm{0}} ^{\mathrm{1}} \boldsymbol{{dx}}.\left[\boldsymbol{{e}}^{\mathrm{2}\boldsymbol{{x}}+\boldsymbol{{y}}} \right]_{\mathrm{0}} ^{\mathrm{1}} \\$$$$=\int_{\mathrm{0}} ^{\mathrm{1}} \left[\boldsymbol{{e}}^{\mathrm{2}\boldsymbol{{x}}+\mathrm{1}} −\boldsymbol{{e}}^{\mathrm{2}\boldsymbol{{x}}} \right]\boldsymbol{{dx}} \\$$$$=\left(\boldsymbol{{e}}−\mathrm{1}\right)\left[\frac{\boldsymbol{{e}}^{\mathrm{2}\boldsymbol{{x}}} }{\mathrm{2}}\right]_{\mathrm{0}\:} ^{\mathrm{1}} \\$$$$=\frac{\mathrm{1}}{\mathrm{2}}\left(\boldsymbol{{e}}−\mathrm{1}\right)\left(\boldsymbol{{e}}^{\mathrm{2}} −\mathrm{1}\right)=\frac{\mathrm{1}}{\mathrm{2}}\left(\boldsymbol{{e}}^{\mathrm{3}} −\boldsymbol{{e}}^{\mathrm{2}} −\boldsymbol{{e}}+\mathrm{1}\right) \\$$

Answered by Ar Brandon last updated on 26/Jun/20

$$\mathcal{I}=\int_{\mathrm{0}} ^{\mathrm{1}} \int_{\mathrm{0}} ^{\mathrm{1}} \mathrm{e}^{\mathrm{2x}+\mathrm{y}} \mathrm{dxdy}=\left[\int_{\mathrm{0}} ^{\mathrm{1}} \mathrm{e}^{\mathrm{2x}} \mathrm{dx}\right]\left[\int_{\mathrm{0}} ^{\mathrm{1}} \mathrm{e}^{\mathrm{y}} \mathrm{dy}\right]=\left[\frac{\mathrm{e}^{\mathrm{2x}} }{\mathrm{2}}\right]_{\mathrm{0}} ^{\mathrm{1}} \centerdot\left[\frac{\mathrm{e}^{\mathrm{y}} }{\mathrm{1}}\right]_{\mathrm{0}} ^{\mathrm{1}} \\$$$$\Rightarrow\mathcal{I}=\left(\frac{\mathrm{e}^{\mathrm{2}} −\mathrm{1}}{\mathrm{2}}\right)\left(\frac{\mathrm{e}−\mathrm{1}}{\mathrm{1}}\right)=\frac{\mathrm{1}}{\mathrm{2}}\left(\mathrm{e}+\mathrm{1}\right)\left(\mathrm{e}−\mathrm{1}\right)^{\mathrm{2}} \\$$

Answered by mathmax by abdo last updated on 26/Jun/20

$$\mathrm{I}\:=\int_{\mathrm{0}} ^{\mathrm{1}} \:\int_{\mathrm{0}} ^{\mathrm{1}} \:\mathrm{e}^{\mathrm{2x}+\mathrm{y}} \:\mathrm{dy}\:\mathrm{dx}\:\Rightarrow\:\mathrm{I}\:=\int_{\mathrm{0}} ^{\mathrm{1}} \:\mathrm{e}^{\mathrm{2x}} \:\mathrm{dx}.\int_{\mathrm{0}} ^{\mathrm{1}} \:\mathrm{e}^{\mathrm{y}} \:\mathrm{dy} \\$$$$=\left[\frac{\mathrm{1}}{\mathrm{2}}\mathrm{e}^{\mathrm{2x}} \right]_{\mathrm{0}} ^{\mathrm{1}} .\left[\mathrm{e}^{\mathrm{y}} \right]_{\mathrm{0}} ^{\mathrm{1}} \:=\frac{\mathrm{1}}{\mathrm{2}}\left(\mathrm{e}^{\mathrm{2}} −\mathrm{1}\right)\left(\mathrm{e}−\mathrm{1}\right)\:=\frac{\mathrm{1}}{\mathrm{2}}\left(\mathrm{e}^{\mathrm{3}} −\mathrm{e}^{\mathrm{2}} −\mathrm{e}+\mathrm{1}\right) \\$$