Question Number 128922 by mnjuly1970 last updated on 11/Jan/21 $$\:\:\:\:\:\:\:\:\:\:\:\:\:…{advanced}\:\:{calculus}… \\ $$$$\:\:\:\:\:{evaluate}\::::\: \\ $$$$\:\:\:\:\:\:\:\Omega=\int_{\mathrm{0}\:} ^{\:\frac{\mathrm{1}}{\mathrm{2}}} \frac{\mathrm{Arctanh}\left({x}\right)}{{x}}{dx}=? \\ $$$$ \\ $$ Answered by Dwaipayan Shikari last…
Question Number 63372 by necx1 last updated on 03/Jul/19 $${For}\:{what}\:{values}\:{of}\:{a}\:{and}\:{b}\:{will}\:{the} \\ $$$${integral}\:\int_{{a}} ^{{b}} \sqrt{\mathrm{10}−{x}−{x}^{\mathrm{2}} }{dx}\:{be}\:{at} \\ $$$${maximum} \\ $$ Commented by mr W last updated…
Question Number 128900 by bemath last updated on 11/Jan/21 $$\:\int\:\frac{\mathrm{4x}+\mathrm{5}}{\left(\mathrm{x}+\mathrm{2}\right)\left(\mathrm{x}+\mathrm{3}\right)\left(\mathrm{x}+\mathrm{4}\right)\left(\mathrm{x}+\mathrm{5}\right)+\mathrm{1}}\:\mathrm{dx}?\: \\ $$ Answered by liberty last updated on 11/Jan/21 $$\:\int\:\frac{\mathrm{4x}+\mathrm{5}}{\left(\mathrm{x}^{\mathrm{2}} +\mathrm{7x}+\mathrm{10}\right)\left(\mathrm{x}^{\mathrm{2}} +\mathrm{7x}+\mathrm{12}\right)+\mathrm{1}}\:\mathrm{dx}=\left(\ast\right) \\ $$$$\:\left(\mathrm{x}^{\mathrm{2}} +\mathrm{7x}+\mathrm{10}\right)\left(\mathrm{x}^{\mathrm{2}}…
Question Number 128888 by bemath last updated on 11/Jan/21 $$\:\int\:\frac{\left(\mathrm{x}−\mathrm{1}\right)\left(\mathrm{x}−\mathrm{2}\right)\left(\mathrm{x}−\mathrm{3}\right)}{\left(\mathrm{x}−\mathrm{4}\right)\left(\mathrm{x}−\mathrm{5}\right)\left(\mathrm{x}−\mathrm{6}\right)}\:\mathrm{dx}\:=? \\ $$ Answered by Olaf last updated on 11/Jan/21 $$\Omega\:=\:\int\frac{\left({x}−\mathrm{1}\right)\left({x}−\mathrm{2}\right)\left({x}−\mathrm{3}\right)}{\left({x}−\mathrm{4}\right)\left({x}−\mathrm{5}\right)\left({x}−\mathrm{6}\right)}{dx} \\ $$$$\Omega\:=\:\int\left(\mathrm{1}+\frac{\mathrm{A}}{{x}−\mathrm{4}}+\frac{\mathrm{B}}{{x}−\mathrm{5}}+\frac{\mathrm{C}}{{x}−\mathrm{6}}\right){dx} \\ $$$$\mathrm{A}\:=\:\frac{\mathrm{3}×\mathrm{2}×\mathrm{1}}{\left(−\mathrm{1}\right)\left(−\mathrm{2}\right)}\:=\:\mathrm{3} \\…
Question Number 63351 by aliesam last updated on 02/Jul/19 Commented by mathmax by abdo last updated on 03/Jul/19 $${let}\:{I}\:=\int\:\:\:\frac{{dx}}{\mathrm{4}+\left({x}−\mathrm{3}\right)^{\mathrm{2}} }\:\:\:{changement}\:\:{x}−\mathrm{3}\:=\mathrm{2}{t}\:{give} \\ $$$${I}\:=\:\int\:\:\frac{\mathrm{2}{dt}}{\mathrm{4}+\mathrm{4}{t}^{\mathrm{2}} }\:=\:\int\:\:\:\:\frac{{dt}}{\mathrm{2}\left(\mathrm{1}+{t}^{\mathrm{2}} \right)}\:=\frac{\mathrm{1}}{\mathrm{2}}\:{arctan}\left({t}\right)+{c} \\…
Question Number 128853 by Ar Brandon last updated on 10/Jan/21 $$\mathrm{F}\left(\mathrm{x}\right)=\int_{\mathrm{x}} ^{\mathrm{2x}} \frac{\mathrm{dx}}{\:\sqrt{\mathrm{t}^{\mathrm{4}} +\mathrm{t}^{\mathrm{2}} +\mathrm{1}}} \\ $$$$\mathrm{1}.\:\mathrm{Show}\:\mathrm{that}\:\mathrm{F}\:\mathrm{is}\:\mathrm{defined},\:\mathrm{continuous}\:\mathrm{and}\:\mathrm{derivable}\:\mathrm{in}\:\mathbb{R} \\ $$ Answered by mathmax by abdo last…
Question Number 128851 by mnjuly1970 last updated on 10/Jan/21 $$\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:…{nice}\:\:\:{calculus}\:\:\left({I}\right)… \\ $$$$\:\:\:{calculate}\:\::: \\ $$$$ \\ $$$$\:\:\:\:\:\Psi=\int_{\mathrm{0}} ^{\:\mathrm{1}} \left\{\left({e}−\mathrm{1}\right)\sqrt{{log}\left(\:\mathrm{1}+{ex}−{x}\:\right)}\:+{e}^{{x}^{\mathrm{2}} } \right\}{dx}=? \\ $$$$ \\ $$ Commented…
Question Number 128841 by mnjuly1970 last updated on 10/Jan/21 $$\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:…{nice}\:{calculus}\:… \\ $$$$\:\:\:\:\mathscr{E}{valuation}\:{of}\:::\:\Phi=\:\int_{\mathrm{0}} ^{\:\mathrm{1}} {ln}\left({x}\right).{arctan}\left({x}\right){dx} \\ $$$${solution}:: \\ $$$$\:\:\:\:{note}\:\mathrm{1}::\:\underset{{n}=\mathrm{1}} {\overset{\infty} {\sum}}\frac{\left(−\mathrm{1}\right)^{{n}−\mathrm{1}} }{\mathrm{2}{n}−\mathrm{1}}={arctan}\left(\mathrm{1}\right)=\frac{\pi}{\mathrm{4}} \\ $$$${note}\:\mathrm{2}\:::\:\underset{{n}=\mathrm{1}} {\overset{\infty} {\sum}}\frac{\left(−\mathrm{1}\right)^{{n}−\mathrm{1}}…
Question Number 63292 by aliesam last updated on 02/Jul/19 $$\int_{\mathrm{0}} ^{\mathrm{1}} \int_{\mathrm{0}} ^{\mathrm{1}} \:\frac{{dy}}{\mathrm{1}+{y}\left({x}^{\mathrm{2}} −{x}\right)}\:{dx} \\ $$ Commented by mathmax by abdo last updated on…
Question Number 128826 by benjo_mathlover last updated on 10/Jan/21 $$\int_{−\mathrm{1}} ^{\:\mathrm{5}} \sqrt{\frac{\mathrm{2x}^{\mathrm{2}} −\mathrm{8}}{\mathrm{x}}}\:\mathrm{dx}\:=? \\ $$ Answered by bobhans last updated on 10/Jan/21 $$\:\int\:\sqrt{\frac{\left({x}\sqrt{\mathrm{2}}\right)^{\mathrm{2}} −\left(\sqrt{\mathrm{8}}\right)^{\mathrm{2}} }{{x}}}\:{dx}\:…