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Category: Integration

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Question Number 38198 by maxmathsup by imad last updated on 22/Jun/18 $${we}\:{give}\:\int_{\mathrm{0}} ^{\infty} \:\:{e}^{−{x}} {ln}\left({x}\right){dx}=−\gamma \\ $$$$\left.\mathrm{1}\right)\:{calculate}\:\:{f}\left({a}\right)=\:\int_{\mathrm{0}} ^{\infty} \:\:{e}^{−{ax}} {ln}\left({x}\right){dx}\:\:{with}\:{a}>\mathrm{0} \\ $$$$\left.\mathrm{2}\right)\:{let}\:{u}_{{n}} =\:\int_{\mathrm{0}} ^{\infty} \:\:{e}^{−{nx}}…

Question-169259

Question Number 169259 by mathlove last updated on 27/Apr/22 Commented by infinityaction last updated on 27/Apr/22 $$\:\:\:\:\:\:\:\:{I}\left({a}\right)\:=\:\int_{\mathrm{0}} ^{\infty} \frac{{e}^{−{ax}} }{{x}}\:{dx}\:\:\:\:\:\:{and}\:\:\:\:{I}\left({b}\right)\:=\:\int_{\mathrm{0}} ^{\infty} \frac{{e}^{−{bx}} }{{x}}{dx} \\ $$$$\:\:\:\:\:\:{I}'\left({a}\right)\:\:=\:\:−\int_{\mathrm{0}}…

Solve-for-n-such-that-1-1-2-1-n-n-1-0-1-x-n-1-1-x-dx-ln2-1-n-1-

Question Number 103723 by Ar Brandon last updated on 16/Jul/20 $$\mathrm{Solve}\:\mathrm{for}\:\mathrm{n},\:\mathrm{such}\:\mathrm{that}; \\ $$$$\mathrm{1}−\frac{\mathrm{1}}{\mathrm{2}}+\centerdot\centerdot\centerdot+\frac{\left(−\mathrm{1}\right)^{\mathrm{n}} }{\mathrm{n}+\mathrm{1}}=\int_{\mathrm{0}} ^{\mathrm{1}} \frac{\mathrm{x}^{\mathrm{n}+\mathrm{1}} }{\mathrm{1}+\mathrm{x}}\mathrm{dx}−\mathrm{ln2}−\left(−\mathrm{1}\right)^{\mathrm{n}+\mathrm{1}} \\ $$ Terms of Service Privacy Policy Contact:…

Question-169230

Question Number 169230 by Giantyusuf last updated on 26/Apr/22 Answered by Mathspace last updated on 26/Apr/22 $${I}=_{{x}=\mathrm{3}^{\frac{\mathrm{1}}{\mathrm{4}}} {t}} \:\:\:\int\:\:\:\:\frac{\sqrt{\mathrm{3}}{t}^{\mathrm{2}} }{\mathrm{3}\left(\mathrm{1}+{t}^{\mathrm{4}} \right)}\mathrm{3}^{\frac{\mathrm{1}}{\mathrm{4}}} {dt} \\ $$$$=\mathrm{3}^{−\frac{\mathrm{1}}{\mathrm{2}}+\frac{\mathrm{1}}{\mathrm{4}}} \:\int\:\:\frac{{t}^{\mathrm{2}}…