Question Number 26756 by abdo imad last updated on 28/Dec/17 $${prove}\:{that}\:\:\int_{\mathrm{0}} ^{\mathrm{1}} \:\:\frac{{dx}}{{x}+\:{e}^{{x}} }\:=\:\sum_{{n}=\mathrm{0}} ^{\propto} \:\:\frac{\left(−\mathrm{1}\right)^{{n}} }{\left({n}+\mathrm{1}\right)^{{n}+\mathrm{1}} }\:{A}_{{n}} \\ $$$${with}\:\:{A}_{{n}} \:=\:\int_{\mathrm{0}} ^{{n}+\mathrm{1}} \:{t}^{{n}} \:{e}^{−{t}} {dt}\:.…
Question Number 26757 by abdo imad last updated on 28/Dec/17 $${give}\:{the}\:{decomposition}\:{of}\:{F}\left({x}\right)\:=\:\:\:\frac{\mathrm{1}}{{x}^{\mathrm{2}{n}} +\mathrm{1}}\:\:{inside}\:\mathbb{C}\left[{x}\right] \\ $$$${then}\:{find}\:{the}\:{value}\:{of}\:\:\int_{\mathrm{0}} ^{\infty} \:\:\frac{{dx}}{\mathrm{1}+{x}^{\mathrm{2}{n}} }\:\:\:\:\:\:{n}\in\mathbb{N}\:\:{and}\:{n}\neq{o} \\ $$ Commented by abdo imad last updated…
Question Number 157826 by cortano last updated on 28/Oct/21 Answered by MJS_new last updated on 29/Oct/21 $$\underset{\mathrm{0}} {\overset{\mathrm{1}} {\int}}\frac{{dx}}{\left(\mathrm{2}+\mathrm{5}{x}\right)\sqrt[{\mathrm{4}}]{\mathrm{2}{x}^{\mathrm{3}} \left(\mathrm{1}−{x}\right)}}= \\ $$$$\:\:\:\:\:\left[{t}=\sqrt[{\mathrm{4}}]{\frac{\mathrm{7}{x}}{\mathrm{2}\left(\mathrm{1}−{x}\right)}}\:\rightarrow\:{dx}=\mathrm{4}\sqrt[{\mathrm{4}}]{\frac{\mathrm{2}{x}^{\mathrm{3}} \left(\mathrm{1}−{x}\right)^{\mathrm{5}} }{\mathrm{7}}}{dt}\right] \\…
Question Number 157823 by mnjuly1970 last updated on 28/Oct/21 $$ \\ $$$$\:\:\:\:\:\:\:{calculate}\:: \\ $$$$\:\:\:\:\Omega\::=\:\underset{{n}=\mathrm{0}} {\overset{\infty} {\sum}}\frac{\:\mathrm{1}}{\left(\mathrm{4}{n}+\mathrm{1}\right)^{\:\mathrm{3}} }\:\:\:=\:? \\ $$$$\: \\ $$ Answered by qaz last…
Question Number 26738 by goswamisubhabrata007@gmail.com last updated on 28/Dec/17 Commented by prakash jain last updated on 28/Dec/17 $$\int{e}^{{x}} \left(\mathrm{log}\:{x}+\frac{\mathrm{1}}{{x}}\right){dx} \\ $$$$\int{e}^{{x}} \left[{f}\left({x}\right)+{f}'\left({x}\right)\right]{dx} \\ $$$$=\int{e}^{{x}} {f}\left({x}\right){dx}+\int{e}^{{x}}…
Question Number 92232 by M±th+et+s last updated on 05/May/20 Commented by M±th+et+s last updated on 05/May/20 $${find}\:\int_{{C}} \frac{{e}^{{z}} −\mathrm{1}}{{z}^{\mathrm{2}} \left({z}−\mathrm{1}\right)}{dz} \\ $$ Terms of Service…
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Question Number 157744 by mnjuly1970 last updated on 27/Oct/21 $$ \\ $$$$\:\:\:\:\int_{\mathrm{0}} ^{\:\mathrm{1}} \frac{\:\:\mathrm{Li}_{\:\mathrm{2}} \:\left(−\mathrm{x}\:\right)\:}{\mathrm{1}+\:\mathrm{x}}\mathrm{dx}=? \\ $$$$ \\ $$$$ \\ $$ Answered by mindispower last…
Question Number 157750 by cortano last updated on 27/Oct/21 $$\:\int_{\mathrm{0}} ^{\infty} \frac{\mathrm{1}−\mathrm{cos}\:\mathrm{4}{x}}{{xe}^{{x}} }\:{dx}=? \\ $$ Answered by qaz last updated on 27/Oct/21 $$\int_{\mathrm{0}} ^{\infty} \frac{\mathrm{1}−\mathrm{cos}\:\mathrm{4x}}{\mathrm{xe}^{\mathrm{x}}…