Question Number 182238 by HeferH last updated on 06/Dec/22 Answered by Acem last updated on 06/Dec/22 $${a}=\mathrm{5}\int_{\mathrm{0}} ^{\:\infty} \:\frac{\mathrm{sin}\:{x}}{{x}}\:{dx}=\:\:\:\frac{\mathrm{5}\pi}{\mathrm{2}} \\ $$$$ \\ $$$${XD} \\ $$…
Question Number 116701 by bemath last updated on 06/Oct/20 $$\int\:\frac{\mathrm{dx}}{\mathrm{x}+\mathrm{x}\sqrt{\mathrm{x}}}\:=? \\ $$ Commented by bobhans last updated on 06/Oct/20 $$\:\int\:\frac{\mathrm{dx}}{\mathrm{x}+\mathrm{x}\sqrt{\mathrm{x}}\:}\:=\:\int\:\frac{\mathrm{dx}}{\mathrm{x}\left(\mathrm{1}+\sqrt{\mathrm{x}}\:\right)} \\ $$$$\left[\:\mathrm{letting}\:\mathrm{x}\:=\:\lambda^{\mathrm{2}} \:\rightarrow\mathrm{dx}\:=\:\mathrm{2}\lambda\:\mathrm{d}\lambda\:\right]\: \\ $$$$\mathrm{I}=\int\:\frac{\mathrm{2}\lambda\:\mathrm{d}\lambda}{\lambda^{\mathrm{2}}…
Question Number 116687 by megrex last updated on 05/Oct/20 $${Help}\:{please},\:{to}\:{solve}\:{this}\:… \\ $$$${If}\:{f}\left({x}\right)=\mathrm{1}+{x}^{\mathrm{2}} \:\:{for}\:{x}\in\left[−\mathrm{2},\mathrm{2}\right]\:{and}\: \\ $$$$\:\:\:\:\:\:{f}\left({x}\right)=\mathrm{5}\:\:\:\:\:\:\:\:{otherwise}. \\ $$$${Then}\:{what}\:{is}\:{the}\:{value}\:{of} \\ $$$$\int_{−\mathrm{2}} ^{+\mathrm{2}} {f}\left(\mathrm{2}{x}^{\mathrm{2}} \right){dx}? \\ $$$$ \\…
Question Number 116672 by mnjuly1970 last updated on 05/Oct/20 $$\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:…\:\:\:{nice}\:\:{calculus}\:… \\ $$$$ \\ $$$$\:\:\:\:\:\:\:\:\:\:\:\:\:{prove}\:\:{that}\:: \\ $$$$ \\ $$$$\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\mathrm{I}\:=\:\int_{\mathrm{0}} ^{\:\mathrm{1}} \left(\frac{\pi}{\mathrm{4}}\:\:−\mathscr{A}{rctan}\left({x}\right)\right)\frac{{dx}}{\mathrm{1}−{x}^{\mathrm{2}} }\:=\:\frac{\mathrm{G}}{\mathrm{2}}\:\:\checkmark\:\:\:\: \\ $$$$\:\:\:\:\:\:\: \\ $$$$\mathrm{G}\:{is}\:\:\:{catalan}\:\:{constant}\:……
Question Number 116667 by mnjuly1970 last updated on 05/Oct/20 $$\:\:\:\:\:\:\:\:\:\:\:\:…\:\:\:{nice}\:\:{calculus}… \\ $$$$ \\ $$$$\:\:\:\:\:\:{very}\:{nice}\:\:{integral}:: \\ $$$$\:\:\:\:\:\:\:{demonstrate}::: \\ $$$$ \\ $$$$\:\:\:\:\:\:\:\:\Omega\:=\:\int_{\mathrm{0}} ^{\:\mathrm{1}} \frac{\mathrm{1}−{x}}{\left(\mathrm{1}+{x}+{x}^{\mathrm{2}} +{x}^{\mathrm{3}} \right){log}\left({x}\right)}\:{dx}\overset{???} {=}{log}\left(\sqrt{\frac{\mathrm{1}}{\mathrm{2}}}\right)…
Question Number 51122 by behi83417@gmail.com last updated on 24/Dec/18 $$\int\:\:\:\:\frac{\boldsymbol{\mathrm{dx}}}{\boldsymbol{\mathrm{tgx}}−\sqrt{\boldsymbol{\mathrm{tgx}}}}=? \\ $$ Commented by MJS last updated on 24/Dec/18 $$\mathrm{long}\:\mathrm{way}… \\ $$$$\mathrm{start}\:\mathrm{with} \\ $$$${t}=\sqrt{\mathrm{tan}\:{x}}\:\rightarrow\:{dx}=\frac{\mathrm{2}{t}}{{t}^{\mathrm{4}} +\mathrm{1}}{dt}…
Question Number 182170 by mathlove last updated on 05/Dec/22 Answered by SEKRET last updated on 05/Dec/22 $$\:\:\frac{\boldsymbol{\pi}^{\mathrm{2}} }{\mathrm{12}} \\ $$$$ \\ $$ Commented by SEKRET…
Question Number 116626 by frc2crc last updated on 05/Oct/20 $$\underset{{n}=\mathrm{1}} {\overset{\infty} {\prod}}\left(\frac{{a}^{\mathrm{2}} {n}^{\mathrm{2}} }{\left.\left({an}\right)^{\mathrm{2}} −\mathrm{1}\right)}\right)=??? \\ $$$$ \\ $$ Answered by mnjuly1970 last updated on…
Question Number 116627 by mnjuly1970 last updated on 05/Oct/20 $$\:\:\:\:\:\:\:\:\:\: \\ $$$$\:\:\:\:\:\:\:\:\:\:\:\:…\:\:{nice}\:\:\:{calculus}… \\ $$$$ \\ $$$$\:\:\:\:\:\:{please}\:\:\:{evaluate}\::: \\ $$$$ \\ $$$$\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\Phi\:=\:\frac{\left(\int_{\mathrm{0}} ^{\:\frac{\pi}{\mathrm{2}}} \frac{{e}^{{x}} +{e}^{−{x}} }{{sin}\left({x}\right)+{cos}\left({x}\right)}{dx}\right)^{\mathrm{2}} }{\left(\int_{\mathrm{0}}…
Question Number 116590 by bemath last updated on 05/Oct/20 $$\:\int\:\frac{\sqrt{\mathrm{x}}}{\mathrm{1}+\mathrm{x}^{\mathrm{3}} }\:\mathrm{dx}\:=? \\ $$ Answered by Olaf last updated on 05/Oct/20 $${u}\:=\:{x}^{\mathrm{3}/\mathrm{2}} ,\:{du}\:=\:\frac{\mathrm{3}}{\mathrm{2}}\sqrt{{x}}{dx}\:=\:\frac{\mathrm{3}}{\mathrm{2}}{u}^{\mathrm{1}/\mathrm{3}} {dx} \\ $$$$\int\frac{{u}^{\mathrm{1}/\mathrm{3}}…