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Category: Integration

find-dx-1-2-1-x-

Question Number 36421 by abdo.msup.com last updated on 01/Jun/18 $${find}\:\int\:\:\:\:\frac{{dx}}{\mathrm{1}+\mathrm{2}\sqrt{\mathrm{1}−{x}}} \\ $$ Commented by abdo.msup.com last updated on 04/Jun/18 $${changement}\:\sqrt{\mathrm{1}−{x}}={t}\:{give}\:\mathrm{1}−{x}={t}^{\mathrm{2}} \\ $$$${I}\:=\:\int\:\:\:\:\:\frac{−\mathrm{2}{tdt}}{\mathrm{1}+\mathrm{2}{t}}\:=\:−\int\:\frac{\mathrm{2}{t}+\mathrm{1}−\mathrm{1}}{\mathrm{2}{t}\:+\mathrm{1}}{dt} \\ $$$$=−{t}\:+\:\int\:\:\:\:\frac{{dt}}{\mathrm{2}{t}\:+\mathrm{1}}\:=\:−{t}\:\:+\frac{\mathrm{1}}{\mathrm{2}}{ln}\mid\mathrm{2}{t}+\mathrm{1}\mid\:+{c} \\…

calculate-I-0-pi-2-x-3-x-cos-2-xdx-and-J-0-pi-2-x-3-x-sin-2-xdx-cslculate-I-and-J-

Question Number 36415 by abdo.msup.com last updated on 01/Jun/18 $${calculate}\:{I}\:=\:\int_{\mathrm{0}} ^{\frac{\pi}{\mathrm{2}}} \left({x}^{\mathrm{3}} \:+{x}\right){cos}^{\mathrm{2}} {xdx}\:{and} \\ $$$${J}\:=\:\int_{\mathrm{0}} ^{\frac{\pi}{\mathrm{2}}} \:\:\left({x}^{\mathrm{3}} \:+{x}\right){sin}^{\mathrm{2}} {xdx} \\ $$$${cslculate}\:{I}\:{and}\:{J}\:. \\ $$ Commented…

let-I-n-0-1-x-n-3-x-dx-1-calculate-lim-n-I-n-2-calculate-lim-n-n-I-n-

Question Number 36413 by abdo.msup.com last updated on 01/Jun/18 $${let}\:\:{I}_{{n}} =\:\int_{\mathrm{0}} ^{\mathrm{1}} \:{x}^{{n}} \sqrt{\mathrm{3}+{x}}{dx} \\ $$$$\left.\mathrm{1}\right){calculate}\:{lim}_{{n}\rightarrow+\infty} {I}_{{n}} \\ $$$$\left.\mathrm{2}\right)\:{calculate}\:{lim}_{{n}\rightarrow+\infty} \:{n}\:{I}_{{n}} \\ $$ Commented by abdo.msup.com…