Question Number 92323 by Power last updated on 06/May/20 Commented by Prithwish Sen 1 last updated on 06/May/20 $$\mathrm{split}\:\boldsymbol{\mathrm{x}}+\mathrm{3}\:\boldsymbol{\mathrm{into}}\:\frac{\mathrm{1}}{\mathrm{8}}\left(\mathrm{8}\boldsymbol{\mathrm{x}}+\mathrm{4}\right)+\frac{\mathrm{5}}{\mathrm{2}} \\ $$ Commented by Power last…
Question Number 26781 by shubhabrata04@gmail.com last updated on 29/Dec/17 Commented by prakash jain last updated on 29/Dec/17 $$\int_{\mathrm{0}} ^{\mathrm{1}} \left(\mathrm{log}\:\left(\mathrm{1}−{x}\right)−\mathrm{log}\:{x}\right){dx} \\ $$$$=\int_{\mathrm{0}} ^{\mathrm{1}} \mathrm{log}\:\left(\mathrm{1}−{x}\right){dx}−\int_{\mathrm{0}} ^{\mathrm{1}}…
Question Number 26758 by abdo imad last updated on 28/Dec/17 $${let}\:{give}\:{D}=\left\{\left(\:\:{x},{y}\:\right)\in\mathbb{R}^{\mathrm{2}} /{x}^{\mathrm{2}} −{x}\:+{y}^{\mathrm{2}} \leqslant\:\mathrm{4}\:{and}\:\:\mathrm{0}\leqslant{y}\leqslant\mathrm{1}\right\} \\ $$$${calculate}\:\int\int_{{D}} {ln}\left({xy}\right)\sqrt{\:{x}^{\mathrm{2}} +{y}^{\mathrm{2}} {dxdy}\:\:} \\ $$ Commented by abdo imad…
Question Number 26759 by abdo imad last updated on 29/Dec/17 $${find}\:\:{the}\:{value}\:{of}\:\:\:\int_{\mathrm{0}} ^{\:\propto} \:\:\frac{{dx}}{\left({x}+\mathrm{1}\right)\left({x}+\mathrm{2}\right)\left({x}+\mathrm{3}\right)}\:. \\ $$ Commented by abdo imad last updated on 30/Dec/17 $${we}\:{use}\:{the}\:{changement}\:{x}+\mathrm{2}={t} \\…
Question Number 26755 by abdo imad last updated on 28/Dec/17 $${find}\:\:\int\:\:\frac{{dx}}{{x}^{\mathrm{6}} −\mathrm{1}}\:\:. \\ $$ Answered by $@ty@m last updated on 29/Dec/17 $$\int\frac{{dx}}{\left({x}^{\mathrm{3}} +\mathrm{1}\right)\left({x}^{\mathrm{3}} −\mathrm{1}\right)} \\…
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}}…