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Category: Integration

let-F-x-x-1-x-arctan-t-t-dt-1-calculate-dF-dx-x-2-find-F-x-

Question Number 36438 by prof Abdo imad last updated on 02/Jun/18 $${let}\:{F}\left({x}\right)\:=\int_{{x}} ^{\frac{\mathrm{1}}{{x}}} \:\:\frac{{arctan}\left({t}\right)}{{t}}{dt} \\ $$$$\left.\mathrm{1}\right)\:{calculate}\:\:\frac{{dF}}{{dx}}\left({x}\right) \\ $$$$\left.\mathrm{2}\right)\:{find}\:{F}\left({x}\right). \\ $$ Commented by tanmay.chaudhury50@gmail.com last updated…

find-sinx-1-cos-3-x-dx-

Question Number 36436 by prof Abdo imad last updated on 02/Jun/18 $${find}\:\:\int\:\:\:\:\:\frac{{sinx}}{\mathrm{1}+{cos}^{\mathrm{3}} {x}}{dx} \\ $$ Answered by tanmay.chaudhury50@gmail.com last updated on 02/Jun/18 $$=\int\frac{−{dt}}{\left(\mathrm{1}+{t}\right)\left(\mathrm{1}−{t}+{t}^{\mathrm{2}} \right)}\:\:\:\:{t}={cosx} \\…

find-the-value-of-h-t-0-1-ln-1-tx-2-with-t-1-2-calculate-0-1-ln-1-x-2-dx-3-calculate-0-1-ln-1-x-2-dx-

Question Number 36435 by prof Abdo imad last updated on 02/Jun/18 $${find}\:{the}\:{value}\:{of}\:{h}\left({t}\right)=\int_{\mathrm{0}} ^{\mathrm{1}} {ln}\left(\mathrm{1}+{tx}^{\mathrm{2}} \right)\:\:{with}\:\mid{t}\mid\leqslant\mathrm{1} \\ $$$$\left.\mathrm{2}\right)\:{calculate}\:\int_{\mathrm{0}} ^{\mathrm{1}} \:\:{ln}\left(\mathrm{1}+{x}^{\mathrm{2}} \right){dx} \\ $$$$\left.\mathrm{3}\right)\:{calculate}\:\int_{\mathrm{0}} ^{\mathrm{1}} {ln}\left(\mathrm{1}−{x}^{\mathrm{2}} \right){dx}…