Category: Integration

Question Number 41762 by math khazana by abdo last updated on 12/Aug/18 $$letf\left(x\right)={\int }_0^1\frac{ln(1+{xt}^2)}{1+t^2}dt$$$$1)findasimpleformoff\left(x\right)$$$$\left.\mathrm{2}\right)\:{calculate}\:\:\int_{\mathrm{0}} ^{\mathrm{1}} \:\:\frac{{ln}\left(\mathrm{1}+{t}^{\mathrm{2}} \right)}{\mathrm{1}+{t}^{\mathrm{2}} }{dt}…

Question Number 107286 by mathmax by abdo last updated on 09/Aug/20 $$\mathrm{let}{\mathrm{f}}_{\mathrm{n}}\left(\mathrm{x}\right)={\mathrm{ne}}^{-\mathrm{nx}}\mathrm{calculate}{\lim }_{\mathrm{n}\to +\mathrm{\infty }}{\int }_0^1{\mathrm{f}}_{\mathrm{n}}\left(\mathrm{x}\right)\mathrm{dx}$$$$\mathrm{and}\:\int_{\mathrm{0}} ^{\mathrm{1}} \mathrm{lim}_{\mathrm{n}\rightarrow+\infty} \mathrm{f}_{\mathrm{n}} \left(\mathrm{x}\right)\mathrm{dx}\:\:\mathrm{is}\:\mathrm{the}\:\mathrm{convergence}\:\mathrm{uniform}\:\mathrm{on}\:\left[\mathrm{0},\mathrm{1}\right]? \…

Question Number 172822 by Mikenice last updated on 01/Jul/22 Answered by Mathspace last updated on 02/Jul/22 $$questionnotcleardoyou$$$$mean\sqrt{x}or\mathrm{\Gamma }\left(x\right)\dots$$ Terms of Service Privacy…

Question Number 107283 by mathmax by abdo last updated on 09/Aug/20 $$\mathrm{f}\mathrm{integrable}\mathrm{continue}\mathrm{on}[\mathrm{a},\mathrm{b}]\mathrm{let}\mathrm{m}=\inf \mathrm{f}\left(\mathrm{x}\right)\mathrm{and}\mathrm{M}=\sup \mathrm{f}\left(\mathrm{x}\right)$$$$(\mathrm{x}\in [\mathrm{a},\mathrm{b}]\mathrm{prove}\mathrm{that}{(\mathrm{b}-\mathrm{a})}^2⩽{\int }_{\mathrm{a}}^{\mathrm{b}}\mathrm{f}\left(\mathrm{x}\right)\mathrm{dx}\times {\int }_{\mathrm{a}}^{\mathrm{b}}\frac{\mathrm{dx}}{\mathrm{f}\left(\mathrm{x}\right)}⩽\frac{{(\mathrm{b}-\mathrm{a})}^2}{4}\frac{{(\mathrm{m}+\mathrm{M})}^2}{\mathrm{mM}}$$ Terms of…

Question Number 172819 by Mikenice last updated on 01/Jul/22 Answered by CElcedricjunior last updated on 02/Jul/22 $$\int {\mathbf{x}}^{\frac{3}{2}}\mathbf{arctan}\left({\mathbf{x}}^{\frac{1}{2}}\right)\mathbf{dx}=\mathbf{k}$$$$\mathbf{posons}){\mathbf{x}}^{\frac{1}{2}}=\mathbf{a}=>\mathbf{dx}=\mathbf{ada}$$$$\boldsymbol{\mathrm{k}}=\int\boldsymbol{\mathrm{a}}^{\mathrm{4}} \boldsymbol{\mathrm{arctan}}\left(\boldsymbol{\mathrm{a}}\right)\boldsymbol{\mathrm{da}}…