Question Number 66816 by mathmax by abdo last updated on 20/Aug/19 $${let}\:{x}>\mathrm{0}\:{and}\:{f}\left({x}\right)=\int_{\mathrm{1}} ^{\mathrm{2}} \left({t}+\mathrm{1}\right)\sqrt{{t}^{\mathrm{2}} −\mathrm{2}{xt}−\mathrm{1}}{dt} \\ $$$$\left.\mathrm{1}\right)\:{find}\:{a}\:{explicit}\:{form}\:{of}\:{f}\left({x}\right) \\ $$$$\left.\mathrm{2}\right)\:{determine}\:{also}\:{g}\left({x}\right)=\int_{\mathrm{1}} ^{\mathrm{2}} \frac{{t}^{\mathrm{2}} \:+{t}}{\:\sqrt{{t}^{\mathrm{2}} −\mathrm{2}{xt}−\mathrm{1}}}{dt} \\ $$$$\left.\mathrm{3}\right){find}\:{the}\:{value}\:{of}\:{integrals}\:\:\int_{\mathrm{1}}…
Question Number 66814 by ~ À ® @ 237 ~ last updated on 20/Aug/19 $${Let}\:{consider}\:{an}\:{integer}\:{serie}\:\left\{{a}_{{n}} {x}^{{n}} \right\}\:{given}\:{by}\:\:{a}_{{n}} \:=\:{H}_{{n}} =\underset{{k}=\mathrm{1}} {\overset{{n}} {\sum}}\frac{\mathrm{1}}{{k}}\: \\ $$$$\left.\mathrm{1}\right)\:{Find}\:{out}\:{the}\:{largest}\:{domain}\:{D}\:{of}\:{convergence}\:{of}\:{that}\:{integer}\:{serie} \\ $$$$\left.\mathrm{2}\right)\:\forall\:{x}\in{D}\:\:,\:{explicit}\:{the}\:{sum}\:{S}\left({x}\right)\:{of}\:{the}\:\left\{{a}_{{n}}…
Question Number 66801 by mathmax by abdo last updated on 19/Aug/19 $${let}\:{f}\left({x}\right)\:=\int_{\mathrm{0}} ^{\mathrm{2}} \sqrt{{x}+{t}^{\mathrm{2}} }{dt}\:\:\:{with}\:{x}\geqslant\mathrm{0} \\ $$$$\left.\mathrm{1}\right)\:{calculate}\:{f}\left({x}\right) \\ $$$$\left.\mathrm{2}\right){calculate}\:{g}\left({x}\right)\:=\int_{\mathrm{0}} ^{\mathrm{2}} \:\frac{{dt}}{\:\sqrt{{x}+{t}^{\mathrm{2}} }} \\ $$$$\left.\mathrm{3}\right){find}\:{the}\:{value}\left[{of}\:\int_{\mathrm{0}} ^{\mathrm{2}}…
Question Number 132333 by liberty last updated on 13/Feb/21 $$\:\mathrm{very}\:\mathrm{nice}\:\mathrm{integral} \\ $$$$\int\:\frac{\mathrm{4x}^{\mathrm{3}} +\mathrm{4x}^{\mathrm{2}} +\mathrm{4x}+\mathrm{3}}{\left(\mathrm{x}^{\mathrm{2}} +\mathrm{1}\right)\left(\mathrm{x}^{\mathrm{2}} +\mathrm{x}+\mathrm{1}\right)^{\mathrm{2}} }\:\mathrm{dx}? \\ $$$$ \\ $$ Answered by EDWIN88 last…
Question Number 66794 by mathmax by abdo last updated on 19/Aug/19 $${calculate}\:\int_{\mathrm{0}} ^{\infty} \:\:\frac{{cos}\left(\mathrm{2}{arctan}\left(\mathrm{2}{x}\right)\right)}{\mathrm{9}+{x}^{\mathrm{2}} }{dx} \\ $$ Commented by mathmax by abdo last updated on…
Question Number 66792 by mathmax by abdo last updated on 19/Aug/19 $${calculate}\:\int_{\mathrm{0}} ^{\mathrm{1}} \:{cos}\left(\mathrm{2}\:{arctan}\left({x}\right)\right){dx} \\ $$ Commented by mathmax by abdo last updated on 20/Aug/19…
Question Number 66795 by mathmax by abdo last updated on 19/Aug/19 $${let}\:{f}\left({x}\right)\:={e}^{−{x}} {ln}\left(\mathrm{1}+{x}^{\mathrm{2}} \right) \\ $$$$\left.\mathrm{1}\right)\:{calculate}\:{f}^{\left({n}\right)} \left(\mathrm{0}\right) \\ $$$$\left.\mathrm{2}\right)\:{developp}\:{f}\:{at}\:{integr}\:{serie} \\ $$ Commented by mathmax by…
Question Number 66790 by mathmax by abdo last updated on 19/Aug/19 $${find}\:\int_{\mathrm{0}} ^{\infty} \:\:\frac{{x}}{{sh}\left({x}\right)}{dx} \\ $$ Terms of Service Privacy Policy Contact: info@tinkutara.com
Question Number 66793 by mathmax by abdo last updated on 19/Aug/19 $${calculate}\:\int_{\mathrm{0}} ^{\mathrm{1}} \:{cos}\left(\mathrm{3}{arctanx}\right){dx} \\ $$ Terms of Service Privacy Policy Contact: info@tinkutara.com
Question Number 132324 by mnjuly1970 last updated on 13/Feb/21 $$\:\:\:\:\:\:\:\:\:\:\:\:….{advanced}\:\:\:{calculus}… \\ $$$$\:\:\:{evaluation}\:: \\ $$$$\:\:\:\:\:\:\boldsymbol{\phi}=\int_{\mathrm{0}\:^{\:\:} } ^{\:\infty} \frac{{ln}\left(\mathrm{1}+{x}\right)}{{x}\left(\mathrm{1}+{x}^{\mathrm{2}} \right)}\:{dx} \\ $$$$\:\:\:\:{solution}: \\ $$$$\:\:\boldsymbol{\phi}=\left[\int_{\mathrm{0}} ^{\:\mathrm{1}} \frac{{ln}\left(\mathrm{1}+{x}\right)}{{x}\left(\mathrm{1}+{x}^{\mathrm{2}} \right)}{dx}=\boldsymbol{\phi}_{\mathrm{1}}…