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Category: Integration

let-I-0-pi-4-e-2t-cos-4-t-dt-and-J-0-pi-4-e-2t-sin-4-tdt-1-calculate-I-J-and-I-J-2-find-the-value-of-I-and-J-

Question Number 66334 by mathmax by abdo last updated on 12/Aug/19 $${let}\:{I}\:=\int_{\mathrm{0}} ^{\frac{\pi}{\mathrm{4}}} \:\:{e}^{−\mathrm{2}{t}} \:{cos}^{\mathrm{4}} {t}\:{dt}\:{and}\:{J}=\int_{\mathrm{0}} ^{\frac{\pi}{\mathrm{4}}} \:{e}^{−\mathrm{2}{t}} \:{sin}^{\mathrm{4}} {tdt} \\ $$$$\left.\mathrm{1}\right){calculate}\:\:{I}+{J}\:{and}\:{I}−{J} \\ $$$$\left.\mathrm{2}\right)\:{find}\:{the}\:{value}\:{of}\:{I}\:{and}\:{J}. \\…

let-A-n-0-1-x-n-1-x-dx-1-calculate-A-0-and-A-1-2-prove-that-n-N-3-2n-A-n-2nA-n-1-3-find-A-n-interms-of-n-

Question Number 66332 by mathmax by abdo last updated on 12/Aug/19 $${let}\:{A}_{{n}} =\int_{\mathrm{0}} ^{\mathrm{1}} \:{x}^{{n}} \sqrt{\mathrm{1}−{x}}{dx} \\ $$$$\left.\mathrm{1}\right){calculate}\:{A}_{\mathrm{0}} \:{and}\:{A}_{\mathrm{1}} \\ $$$$\left.\mathrm{2}\right){prove}\:{that}\:\forall{n}\in{N}^{\bigstar} \:\:\:\:\left(\mathrm{3}+\mathrm{2}{n}\right){A}_{{n}} =\mathrm{2}{nA}_{{n}−\mathrm{1}} \\ $$$$\left.\mathrm{3}\right)\:{find}\:{A}_{{n}}…