Question Number 137696 by Mathspace last updated on 05/Apr/21 $${let}\:{U}_{{n}} =\int_{\mathrm{0}} ^{\infty} \:\frac{{cos}\left({nsinx}\right)−{sin}\left({ncosx}\right)}{\left({x}^{\mathrm{2}} \:+\mathrm{3}\right)^{\mathrm{2}} }{dx} \\ $$$${determine}\:{lim}_{{n}\rightarrow+\infty} {U}_{{n}} \\ $$$${and}\:{lim}_{{n}\rightarrow+\infty} {e}^{−{n}^{\mathrm{2}} } \:{U}_{{n}} \\ $$…
Question Number 137695 by Mathspace last updated on 05/Apr/21 $${find}\:\:\int\:\:\frac{\left(\mathrm{2}{x}+\mathrm{1}\right)^{\mathrm{3}} }{\:\sqrt{{x}}+\sqrt{{x}+\mathrm{1}}}{dx} \\ $$ Answered by bemath last updated on 05/Apr/21 $$\:\int\:\frac{\left(\mathrm{2}{x}+\mathrm{1}\right)^{\mathrm{3}} \left(\sqrt{{x}}−\sqrt{{x}+\mathrm{1}}\:\right)}{{x}−\left({x}+\mathrm{1}\right)}{dx} \\ $$$$=\:\int\:\left(\mathrm{2}{x}+\mathrm{1}\right)^{\mathrm{3}} \left(\sqrt{{x}+\mathrm{1}}\:−\sqrt{{x}}\:\right){dx}…
Question Number 137693 by Mathspace last updated on 05/Apr/21 $${calculate}\:\int_{\mathrm{0}} ^{\infty} \:\frac{{cos}\left({sinx}\right)−{sin}\left({cosx}\right)}{{x}^{\mathrm{2}} \:+\mathrm{8}}{dx} \\ $$ Answered by mathmax by abdo last updated on 07/Apr/21 $$\int_{\mathrm{0}}…
Question Number 137694 by Mathspace last updated on 05/Apr/21 $${find}\:\int\:\:\:\frac{{x}^{\mathrm{3}} }{\left({x}−\mathrm{3}\right)^{\mathrm{3}} \left({x}\sqrt{\mathrm{2}}+\mathrm{7}\right)^{\mathrm{4}} }{dx} \\ $$ Commented by MJS_new last updated on 05/Apr/21 $$\mathrm{you}\:\mathrm{can}\:\mathrm{either}\:\mathrm{decompose}\:\mathrm{or}\:\mathrm{apply}\:\mathrm{Ostrogradski}'\mathrm{s} \\ $$$$\mathrm{Method};\:\mathrm{it}'\mathrm{s}\:\mathrm{no}\:\mathrm{fun}\:\mathrm{typing}\:\mathrm{with}\:\mathrm{these}\:\mathrm{constants}……
Question Number 137668 by mnjuly1970 last updated on 05/Apr/21 $$\:\:\:\:\:\:\:\:\:\:\:\:\:\:…….{advanced}\:…\:…\:…\:…\:{calculus}…. \\ $$$$\:\:{prove}\:{that}:::::\:\ast\ast\ast\ast\ast \\ $$$$\:\:\:\:\boldsymbol{\Omega}=\underset{{n}=\mathrm{0}} {\overset{\infty} {\sum}}\frac{\Gamma\left({n}+\mathrm{1}\right)\Gamma\left({x}+\mathrm{1}\right)}{\left({n}+\mathrm{1}\right)\Gamma\left({n}+{x}+\mathrm{2}\right)}=\psi'\left({x}+\mathrm{1}\right) \\ $$$${proof}:: \\ $$$$\:\:\:\:\Omega=\underset{{n}=\mathrm{0}\:} {\overset{\infty} {\sum}}\frac{\beta\:\left({n}+\mathrm{1},{x}+\mathrm{1}\right)}{{n}+\mathrm{1}}=\underset{{n}=\mathrm{0}} {\overset{\infty} {\sum}}\left\{\frac{\mathrm{1}}{\left({n}+\mathrm{1}\right)}\int_{\mathrm{0}} ^{\:\mathrm{1}}…
Question Number 6588 by Temp last updated on 04/Jul/16 $$\mathrm{Expansion}\:\mathrm{of}\:\mathrm{Q6582} \\ $$$$\int_{\mathrm{0}} ^{\:\infty} {e}^{−{ix}^{\mathrm{2}} } {dx}=??? \\ $$ Commented by FilupSmith last updated on 07/Jul/16…
Question Number 137656 by rs4089 last updated on 05/Apr/21 $$\int_{\mathrm{0}} ^{\infty} {sin}\left({x}^{{p}} \right){dx} \\ $$ Answered by Dwaipayan Shikari last updated on 05/Apr/21 $$\int_{\mathrm{0}} ^{\infty}…
Question Number 6582 by Temp last updated on 04/Jul/16 $$\int{e}^{−{ix}^{\mathrm{2}} } {dx}=?? \\ $$ Answered by Yozzii last updated on 04/Jul/16 $${e}^{{u}} =\underset{{r}=\mathrm{0}} {\overset{\infty} {\sum}}\frac{{u}^{{r}}…
Question Number 137605 by mnjuly1970 last updated on 04/Apr/21 $$\:\:\:\:\:\:\:\:\:\:\:\:…..{nice}\:\:\:\:{calculus}… \\ $$$$\:\:\:\:\:{prove}\:{that}:: \\ $$$$\:\:\:\:\:\underset{{n}=\mathrm{0}} {\overset{\infty} {\sum}}{tan}^{−\mathrm{1}} \left(\frac{\mathrm{1}}{{F}_{{n}} }\right).{tan}^{−\mathrm{1}} \left(\frac{\mathrm{1}}{{F}_{{n}+\mathrm{1}} }\right)=\frac{\pi^{\mathrm{2}} }{\mathrm{4}} \\ $$$$\:\:{F}_{{n}} \:{is}\:{fibonacci}\:{sequence}…. \\…
Question Number 137610 by mnjuly1970 last updated on 04/Apr/21 $$\:\:\:\:\:\:\:\:\:\:……..\:{mathematical}\:\:\:{analysis}\:\left({II}\right)…. \\ $$$$\:\:\:\:{prove}\:\:{that}\::: \\ $$$$\:\:\:\:\:\boldsymbol{\Omega}=\int_{\mathrm{0}} ^{\:\mathrm{1}} \frac{\mathrm{1}}{\mathrm{1}+{x}}{ln}\left(\frac{{x}^{\mathrm{2}} +\mathrm{2}{x}+\mathrm{1}}{\mathrm{1}+{x}+{x}^{\mathrm{2}} }\right)=\underset{{n}=\mathrm{1}} {\overset{\infty} {\sum}}\frac{\mathrm{1}}{{n}^{\mathrm{2}} \begin{pmatrix}{\mathrm{2}{n}}\\{\:\:{n}}\end{pmatrix}}=\frac{\pi^{\mathrm{2}} }{\mathrm{18}}.. \\ $$ Terms…