Question Number 62437 by mathsolverby Abdo last updated on 21/Jun/19 $${let}\:{f}\left({x}\right)\:=\int_{\mathrm{0}} ^{\mathrm{1}} \:\frac{{arctan}\left(\mathrm{1}+{xt}\right)}{{t}^{\mathrm{2}} \:+\mathrm{1}}{dt} \\ $$$${determine}\:{a}\:{explicit}\:{form}\:{for}\:{f}\left({x}\right) \\ $$$$\left.\mathrm{2}\right){calculate}\:\int_{\mathrm{0}} ^{\mathrm{1}} \:\frac{{arctan}\left(\mathrm{1}+\mathrm{2}{t}\right)}{\mathrm{1}+{t}^{\mathrm{2}} }{dt} \\ $$$$ \\ $$…
Question Number 62425 by mathmax by abdo last updated on 21/Jun/19 $${let}\:\xi\left({x}\right)\:=\sum_{{n}=\mathrm{1}} ^{\infty} \:\frac{\mathrm{1}}{{n}^{{x}} }\:\:\:\:\:{with}\:{x}>\mathrm{1} \\ $$$$\left.\mathrm{1}\right)\:{calculate}\:{lim}_{{x}\rightarrow\mathrm{1}^{+} } \:\:\xi\left({x}\right)\:\:{and}\:{lim}_{{x}\rightarrow+\infty} \:\:\:\xi\left({x}\right) \\ $$$$\left.\mathrm{2}\right)\:{prove}\:{that}\:\xi\left({x}\right)\:=\mathrm{1}+\mathrm{2}^{−{x}} \:+{o}\left(\mathrm{2}^{−{x}} \right)\:\:\:\left({x}\rightarrow+\infty\right) \\…
Question Number 62420 by mathsolverby Abdo last updated on 20/Jun/19 $${let}\:{u}_{{n}} \left({x}\right)=\frac{\mathrm{1}}{{n}^{{x}} }\:−\int_{{n}} ^{{n}+\mathrm{1}} \frac{{dt}}{{t}^{{x}} }\:\:{with}\:{x}\in\left[\mathrm{1},\mathrm{2}\right] \\ $$$$\left.\mathrm{1}\right){prove}\:{that}\:\mathrm{0}\leqslant\:{u}_{{n}} \left({x}\right)\leqslant\frac{\mathrm{1}}{{n}^{{x}} }−\frac{\mathrm{1}}{\left({n}+\mathrm{1}\right)^{{x}} }\:\left({n}>\mathrm{0}\right) \\ $$$$\left.\mathrm{2}\right){prove}\:{that}\:\Sigma\:{u}_{{n}} \left({x}\right){converges} \\…
Question Number 127958 by mnjuly1970 last updated on 03/Jan/21 $$\:\:\:\:\:\:\:\:\:\:\:…{nice}\:\:{calculus}… \\ $$$$\:\:\:{calculate} \\ $$$$\:\:\:\:\Omega=\int_{\mathrm{1}} ^{\:\infty} \frac{{ln}\left({x}^{\mathrm{4}} −\mathrm{2}{x}^{\mathrm{2}} +\mathrm{2}\right)}{{x}\sqrt{{x}^{\mathrm{2}} −\mathrm{1}}\:}\:{dx}=? \\ $$$$ \\ $$ Answered by…
Question Number 62418 by mathmax by abdo last updated on 20/Jun/19 $${calculate}\:\int_{\mathrm{0}} ^{\mathrm{1}} \:\Gamma\left({t}\right).\Gamma\left(\mathrm{1}−{t}\right){dt}\: \\ $$ Terms of Service Privacy Policy Contact: info@tinkutara.com
Question Number 62419 by mathmax by abdo last updated on 20/Jun/19 $${calculate}\:\int_{\mathrm{0}} ^{\mathrm{1}} \left(\mathrm{2}{x}^{\mathrm{2}} −\mathrm{1}\right)\sqrt{{x}^{\mathrm{2}} −\mathrm{2}{x}+\mathrm{5}}{dx} \\ $$ Commented by mathmax by abdo last updated…
Question Number 127952 by rs4089 last updated on 03/Jan/21 Answered by Ar Brandon last updated on 03/Jan/21 $$\Psi=\int_{\mathrm{0}} ^{\mathrm{1}} \mathrm{x}^{\mathrm{2}} \sqrt{\frac{\mathrm{1}−\mathrm{x}^{\mathrm{2}} }{\mathrm{1}+\mathrm{x}^{\mathrm{2}} }}\mathrm{dx}=\int_{\mathrm{0}} ^{\mathrm{1}} \left\{\frac{\mathrm{x}^{\mathrm{2}}…
Question Number 62416 by mathmax by abdo last updated on 20/Jun/19 $${let}\:\Gamma\left({x}\right)=\int_{\mathrm{0}} ^{\infty} \:{t}^{{x}−\mathrm{1}} {e}^{−{t}} \:{dt}\:\:\:{with}\:{x}>\mathrm{1}\:{calculate}\:\Gamma^{\left({n}\right)} \left({x}\right)\:{for}\:{all}\:{integr}\:{n}. \\ $$ Commented by mathmax by abdo last…
Question Number 62417 by mathmax by abdo last updated on 20/Jun/19 $${prove}\:{that}\:\Gamma\left({x}\right).\Gamma\left(\mathrm{1}−{x}\right)\:=\frac{\pi}{{sin}\left(\pi{x}\right)}\:\:\:\:\:\:\:{with}\:\mathrm{0}<{x}<\mathrm{1} \\ $$ Terms of Service Privacy Policy Contact: info@tinkutara.com
Question Number 62414 by mathmax by abdo last updated on 20/Jun/19 $${find}\:\int\:\:\:\:\:\frac{{e}^{{x}} }{\:\sqrt{{e}^{\mathrm{2}{x}} −\mathrm{1}}}{dx} \\ $$ Answered by $@ty@m last updated on 20/Jun/19 $${Put}\:{e}^{{x}} ={t}…