Question Number 48715 by Abdo msup. last updated on 27/Nov/18 $$\left.\mathrm{1}\right)\:{find}\:\:{f}\left(\lambda\right)\:=\int_{\mathrm{0}} ^{\mathrm{1}} \:\:\frac{{dx}}{\mathrm{2}+{e}^{−\lambda{x}} }\:\:{with}\:\lambda>\mathrm{0} \\ $$$$\left.\mathrm{2}\right){calculate}\:\int_{\mathrm{0}} ^{\mathrm{1}} \:\:\:\:\frac{{x}}{\left(\mathrm{2}+{e}^{−\lambda{x}} \right)^{\mathrm{2}} }{dx} \\ $$$$\left.\mathrm{3}\right)\:{find}\:{the}\:{value}\:{of}\:\int_{\mathrm{0}} ^{\mathrm{1}} \:\:\:\:\frac{{dx}}{\mathrm{2}\:+{e}^{−{x}\sqrt{\mathrm{3}}} }{dx}\:{and}\:\int_{\mathrm{0}}…
Question Number 48703 by cesar.marval.larez@gmail.com last updated on 27/Nov/18 Answered by tanmay.chaudhury50@gmail.com last updated on 27/Nov/18 $$\left.\mathrm{3}\right)\int{sin}^{\mathrm{3}} \left(\mathrm{2}{x}\right){cos}\left(\mathrm{2}{x}\right){dx} \\ $$$${t}={sin}\mathrm{2}{x}\:\:\:{dt}=\mathrm{2}{cos}\mathrm{2}{xdx} \\ $$$$=\frac{\mathrm{1}}{\mathrm{2}}\int{t}^{\mathrm{3}} {dt} \\ $$$$=\frac{\mathrm{1}}{\mathrm{2}}×\frac{{t}^{\mathrm{4}}…
Question Number 48667 by maxmathsup by imad last updated on 26/Nov/18 $${find}\:\int_{\mathrm{0}} ^{\mathrm{1}} \:\:\frac{{arctan}\left({x}\right)}{\:\sqrt{\mathrm{1}+{x}^{\mathrm{2}} }}\:{dx}\:. \\ $$ Commented by Abdo msup. last updated on 02/Dec/18…
Question Number 48643 by cesar.marval.larez@gmail.com last updated on 26/Nov/18 Commented by maxmathsup by imad last updated on 26/Nov/18 $$\left.\mathrm{1}\right){let}\:\:{I}\:=\int\:\:\frac{\mathrm{3}{x}+\mathrm{2}}{\left({x}−\mathrm{1}\right)\left({x}−\mathrm{2}\right)\left({x}^{\mathrm{2}} \:+\mathrm{3}\right)}{dx}\:{let}\:{decompose}\:{F}\left({x}\right)=\frac{\mathrm{3}{x}+\mathrm{2}}{\left({x}−\mathrm{1}\right)\left({x}−\mathrm{2}\right)\left({x}^{\mathrm{2}} \:+\mathrm{3}\right)} \\ $$$${F}\left({x}\right)\:=\frac{{a}}{{x}−\mathrm{1}}\:+\frac{{b}}{{x}−\mathrm{2}}\:+\frac{{cx}+{d}}{{x}^{\mathrm{2}\:} +\mathrm{3}} \\…
Question Number 114161 by bemath last updated on 17/Sep/20 $$\:\int\:\frac{{dx}}{{x}^{\mathrm{4}} −\mathrm{5}{x}^{\mathrm{2}} −\mathrm{16}} \\ $$ Answered by Olaf last updated on 17/Sep/20 $$−\frac{\mathrm{2}}{\:\sqrt{\mathrm{89}}}\frac{\mathrm{arctan}\left(\frac{\sqrt{\mathrm{2}}{x}}{\:\sqrt{\sqrt{\mathrm{89}}−\mathrm{5}}}\right)}{\:\sqrt{\sqrt{\mathrm{89}}−\mathrm{5}}}−\frac{\mathrm{2}}{\:\sqrt{\mathrm{89}}}\frac{\mathrm{arctan}\left(\frac{\sqrt{\mathrm{2}}{x}}{\:\sqrt{\sqrt{\mathrm{89}}+\mathrm{5}}}\right)}{\:\sqrt{\sqrt{\mathrm{89}}+\mathrm{5}}} \\ $$ Answered…
Question Number 114146 by Eric002 last updated on 17/Sep/20 $${prove} \\ $$$$\int_{\mathrm{0}} ^{\mathrm{1}} \frac{{t}^{{n}+\mathrm{2}} \phi\left({t},\mathrm{1},{n}+\mathrm{2}\right)+{ln}\left(\mathrm{1}−{t}\right)+{t}\:{H}_{{n}+\mathrm{1}} }{{t}\left({t}−\mathrm{1}\right)}{dt} \\ $$$$=\frac{{H}_{{n}+\mathrm{1}} ^{\left(\mathrm{2}\right)} −\left({H}_{{n}} \right)^{\mathrm{2}} }{\mathrm{2}} \\ $$ Commented…
Question Number 114135 by mnjuly1970 last updated on 17/Sep/20 $$\:\:\:\:\:\:\:\:\:\:\:\:….\mathscr{A}{dvanced}\:\:{mathematics}\:… \\ $$$$\:\:\:\:\:\:\:\:\:\:\:\:\:{i}::\:{prove}\:\:{that}\::\:\:\:\:\Omega=\frac{\mathrm{1}}{\pi}\int_{\mathrm{0}} ^{\:\infty} \frac{\mathrm{1}}{\left({x}^{\mathrm{2}} −{x}+\mathrm{1}\right)^{\mathrm{2}} \sqrt{{x}}}{dx}\:=\mathrm{1}\:\: \\ $$$$\:\:\:\:\:\:\:\:\:\:\:\:\:{ii}::{evaluate}\:::\:\:\Phi\:=\:\int_{\mathrm{0}} ^{\:\mathrm{1}} {x}^{\mathrm{2}} \:{ln}\left({x}\right)\:{ln}\left(\mathrm{1}−{x}\right){dx}=??? \\ $$$$\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:….{m}.{n}.{july}.\:\mathrm{1970}…. \\ $$$$\:…
Question Number 179655 by Acem last updated on 31/Oct/22 $${Evaluate}\:\int\mathrm{6}\:\mathrm{arctan}\:\frac{\mathrm{8}}{{w}}\:{dw} \\ $$ Answered by CElcedricjunior last updated on 05/Nov/22 $$\int\mathrm{6}\boldsymbol{{arctan}}\left(\frac{\mathrm{8}}{\boldsymbol{\omega}}\right)\boldsymbol{{d}\omega}=\boldsymbol{{k}} \\ $$$$\boldsymbol{{posons}}\:\begin{cases}{\boldsymbol{{u}}=\boldsymbol{{arctan}}\left(\frac{\mathrm{8}}{\boldsymbol{\omega}}\right)}\\{\boldsymbol{{v}}'=\mathrm{1}}\end{cases}=>\begin{cases}{\boldsymbol{{u}}'=−\frac{\mathrm{8}}{\boldsymbol{\omega}^{\mathrm{2}} }\left(\frac{\mathrm{1}}{\mathrm{1}+\left(\frac{\mathrm{8}}{\boldsymbol{\omega}}\right)^{\mathrm{2}} }\:\right)}\\{\boldsymbol{{v}}=\boldsymbol{\omega}}\end{cases} \\…
Question Number 114105 by gab last updated on 17/Sep/20 $$\int\:\frac{{arctan}\left({e}^{{x}} \right)}{\:\sqrt{{x}}}\:{dx} \\ $$ Commented by sherzodbek last updated on 17/Sep/20 salom Terms of Service Privacy…
Question Number 114102 by bemath last updated on 17/Sep/20 $$\int\:\frac{{dx}}{\mathrm{tan}\:{x}−\mathrm{sin}\:{x}} \\ $$ Answered by bobhans last updated on 17/Sep/20 $$\int\:\frac{{dx}}{\mathrm{tan}\:{x}−\mathrm{sin}\:{x}}\:=\:−\frac{\mathrm{1}}{\mathrm{2}}\mathrm{cosec}\:^{\mathrm{2}} {x}\:−\frac{\mathrm{1}}{\mathrm{2}}\mathrm{cot}\:{x}\:\mathrm{cosec}\:{x}\: \\ $$$$\:\:\:\:\:\:\:\:\:\:\:\:+\:\frac{\mathrm{1}}{\mathrm{2}}\mathrm{ln}\:\mid\mathrm{cot}\:{x}+\mathrm{cosec}\:{x}\mid\:+\:{c} \\ $$…