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Category: Integration

let-I-n-0-1-2-1-2t-n-e-t-dt-with-n-integr-not-0-1-prove-that-t-0-1-2-1-e-1-2t-n-1-2t-n-e-t-1-2t-n-then-find-lim-n-I-n-2-prove-that-I-n-1-1-2-n-1-I-n

Question Number 30764 by abdo imad last updated on 25/Feb/18 $${let}\:\:{I}_{{n}} =\:\int_{\mathrm{0}} ^{\frac{\mathrm{1}}{\mathrm{2}}} \:\left(\mathrm{1}−\mathrm{2}{t}\right)^{{n}} \:{e}^{−{t}} {dt}\:\:{with}\:{n}\:{integr}\:{not}\:\mathrm{0} \\ $$$$\left.\mathrm{1}\right)\:{prove}\:{that}\:\forall{t}\in\left[\mathrm{0},\frac{\mathrm{1}}{\mathrm{2}}\right] \\ $$$$\frac{\mathrm{1}}{\:\sqrt{{e}}}\left(\mathrm{1}−\mathrm{2}{t}\right)^{{n}} \leqslant\:\left(\mathrm{1}−\mathrm{2}{t}\right)^{{n}} \:{e}^{−{t}} \:\leqslant\left(\mathrm{1}−\mathrm{2}{t}\right)^{{n}} \:{then}\:{find}\:{lim}_{{n}\rightarrow\infty\:} {I}_{{n}}…

dx-1-x-14-

Question Number 161830 by EnterUsername last updated on 22/Dec/21 $$\int\frac{{dx}}{\:\sqrt{\mathrm{1}−{x}^{\mathrm{14}} }} \\ $$ Answered by Ar Brandon last updated on 22/Dec/21 $$\mathrm{arcsin}\left({x}\right)=\underset{{n}=\mathrm{0}} {\overset{\infty} {\sum}}\frac{{x}^{\mathrm{2}{n}+\mathrm{1}} \left(\frac{\mathrm{1}}{\mathrm{2}}\right)_{{n}}…

Question-96280

Question Number 96280 by bobhans last updated on 31/May/20 Commented by bemath last updated on 31/May/20 $$\mathrm{let}\:\mathrm{w}\:=\:\mathrm{arctan}\:\mathrm{x} \\ $$$$\mathrm{dw}\:=\:\frac{\mathrm{dx}}{\mathrm{1}+\mathrm{x}^{\mathrm{2}} }\:\Rightarrow\:\int\:\mathrm{w}\:\mathrm{dw}\:=\:\left[\frac{\mathrm{1}}{\mathrm{2}}\mathrm{w}^{\mathrm{2}} \:\right]_{\mathrm{0}} ^{\frac{\pi}{\mathrm{2}}} \\ $$$$=\:\frac{\mathrm{1}}{\mathrm{2}}\left(\frac{\pi}{\mathrm{2}}\right)^{\mathrm{2}} =\:\frac{\pi^{\mathrm{2}}…

let-give-D-R-2-0-0-and-from-R-let-C-1-x-y-D-0-lt-x-2-y-2-1-C-2-x-y-D-x-2-y-2-1-study-the-convergence-of-I-C-1-dxdy-x-2-y-2-and-J-C-2-

Question Number 30741 by abdo imad last updated on 25/Feb/18 $${let}\:{give}\:{D}=\:{R}_{+} ^{\mathrm{2}} \:−\left\{\left(\mathrm{0},\mathrm{0}\right)\right\}\:{and}\:\alpha\:{from}\:{R}\:{let} \\ $$$${C}_{\mathrm{1}} =\left\{\left({x},{y}\right)\in\:{D}/\mathrm{0}<{x}^{\mathrm{2}} \:+{y}^{\mathrm{2}} \leqslant\mathrm{1}\:\right\} \\ $$$${C}_{\mathrm{2}} \:=\left\{\left({x},{y}\right)\:\in{D}\:/\:{x}^{\mathrm{2}} \:+{y}^{\mathrm{2}} \geqslant\mathrm{1}\right\}\:{study}\:{the}\:{convergence}\:{of} \\ $$$${I}=\:\int\int_{{C}_{\mathrm{1}}…

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

Question Number 96257 by bobhans last updated on 31/May/20 $$\int\:{x}^{\mathrm{3}} \:\sqrt{\mathrm{1}−{x}^{\mathrm{2}} }\:{dx}\:?\: \\ $$ Answered by john santu last updated on 31/May/20 $$\int\:{x}^{\mathrm{2}} \:\left({x}\sqrt{\mathrm{1}−{x}^{\mathrm{2}} }\:\right)\:{dx}\:=\:{J}…