Question Number 78280 by msup trace by abdo last updated on 15/Jan/20 $${find}\:{by}\:{recurrence} \\ $$$${J}_{{n},{p}} \:=\int_{\mathrm{0}} ^{\mathrm{1}} {x}^{{n}} \left({arctanx}\right)^{{p}} {dx} \\ $$$${stydy}\:{the}\:{serie}\:\sum_{{n}\geqslant\mathrm{0}\:{and}\:{p}\geqslant\mathrm{0}} \:\:{J}_{{n},{p}} \\ $$…
Question Number 78281 by msup trace by abdo last updated on 15/Jan/20 $${calculate}\:\:\int\int_{{W}} \:\frac{{x}^{\mathrm{2}} −\mathrm{3}{y}^{\mathrm{2}} }{{e}^{{x}^{\mathrm{2}} +{y}^{\mathrm{2}} } }{dxdy} \\ $$$${with}\:{W}\:=\left[\mathrm{0},\mathrm{1}\right]×\left[\mathrm{0},\mathrm{1}\right] \\ $$ Terms of…
Question Number 12744 by tawa last updated on 30/Apr/17 $$\int_{\:\mathrm{e}^{−\mathrm{3}} } ^{\:\mathrm{e}^{−\mathrm{2}} } \:\:\frac{\mathrm{1}}{\left(\mathrm{x}\right)\mathrm{log}\left(\mathrm{x}\right)}\:\mathrm{dx}\:\:=\:\:? \\ $$ Answered by b.e.h.i.8.3.4.1.7@gmail.com last updated on 30/Apr/17 $${logx}={t}\Rightarrow{dx}/{x}={dt} \\…
Question Number 12743 by tawa last updated on 30/Apr/17 $$\mathrm{What}\:\mathrm{is}\:\mathrm{the}\:\mathrm{area}\:\mathrm{of}\:\mathrm{eqilateral}\:\mathrm{triangle}\:\mathrm{whose}\:\mathrm{inscribed}\:\mathrm{circle}\:\mathrm{has}\:\mathrm{a}\:\mathrm{radius}\:\mathrm{2} \\ $$ Answered by mrW1 last updated on 30/Apr/17 $${OC}=\mathrm{2} \\ $$$${OB}=\mathrm{4}={DO} \\ $$$${CB}=\mathrm{2}\sqrt{\mathrm{3}} \\…
Question Number 143812 by liberty last updated on 18/Jun/21 $$\:\mathrm{log}\:_{\mathrm{a}} \left(\mathrm{ax}\right).\mathrm{log}\:_{\mathrm{x}} \left(\mathrm{ax}\right)=\mathrm{log}\:_{\mathrm{a}^{\mathrm{2}} } \left(\frac{\mathrm{1}}{\mathrm{a}}\right) \\ $$$$\:\mathrm{a}>\mathrm{0}\:,\:\mathrm{a}\neq\mathrm{1}\:.\:\mathrm{So}\:\mathrm{x}\:=\:? \\ $$ Answered by Olaf_Thorendsen last updated on 18/Jun/21…
Question Number 12742 by Joel577 last updated on 30/Apr/17 $$\mathrm{Prove}\:\mathrm{that} \\ $$$$\int\:\frac{{dx}}{\left({x}\:+\mathrm{1}\right)^{\mathrm{2}} \:\sqrt{{x}^{\mathrm{2}} \:+\:\mathrm{2}{x}\:+\mathrm{2}}}\:=\:\frac{−\sqrt{{x}^{\mathrm{2}} \:+\:\mathrm{2}{x}\:+\:\mathrm{2}}}{{x}\:+\:\mathrm{1}}\:+\:{C} \\ $$ Answered by b.e.h.i.8.3.4.1.7@gmail.com last updated on 30/Apr/17 $$\frac{\mathrm{1}}{{x}+\mathrm{1}}={t}\Rightarrow\frac{{dx}}{\left({x}+\mathrm{1}\right)^{\mathrm{2}}…
Question Number 78276 by msup trace by abdo last updated on 15/Jan/20 $${find}\:{I}_{{n}} =\int\int_{\left[\mathrm{1},{n}\right]^{\mathrm{2}} } \:\:\:\sqrt{{x}^{\mathrm{2}} +{y}^{\mathrm{2}} }{ln}\left({x}^{\mathrm{2}} +{y}^{\mathrm{2}} \right){dxdy} \\ $$ Commented by mathmax…
Question Number 143814 by mathdanisur last updated on 18/Jun/21 $$\forall{a};{b};{c}\in\mathbb{R}\:,\:{find}\:{all}\:{f}:\mathbb{R}\rightarrow\mathbb{R}\:,\:{such}\:{that} \\ $$$${f}\left({a}\right){f}\left({bc}\right)+\mathrm{9}\leqslant{f}\left({ab}\right)+\mathrm{5}{f}\left({ac}\right) \\ $$ Answered by Olaf_Thorendsen last updated on 18/Jun/21 $${f}\left({a}\right){f}\left({bc}\right)+\mathrm{9}\:\leqslant\:{f}\left({ab}\right)+\mathrm{5}{f}\left({ac}\right) \\ $$$$ \\…
Question Number 78277 by msup trace by abdo last updated on 15/Jan/20 $${calculate}\:\int\int_{{W}} \:\:\:\frac{{e}^{−{x}^{\mathrm{2}} −{y}^{\mathrm{2}} } }{\mathrm{2}\sqrt{{x}^{\mathrm{2}} +{y}^{\mathrm{2}} }+\mathrm{3}}{dxdy} \\ $$$${with}\:{W}\:=\left\{\:\left({x},{y}\right)/\:{x}>\mathrm{0}\:{and}\:{y}>\mathrm{0}\right\} \\ $$ Terms of…
Question Number 12740 by malwaan last updated on 30/Apr/17 $$\int\mid\mathrm{x}\mid\:\mathrm{dx} \\ $$ Answered by b.e.h.i.8.3.4.1.7@gmail.com last updated on 30/Apr/17 $$\mid{x}\mid=\begin{cases}{{x}\:\:{if}\:\:{x}\geqslant\mathrm{0}.}\\{−{x}\:{if}\:\:{x}<\mathrm{0}}\end{cases} \\ $$$${I}=\left(\int{xdx}\right)\:{or}\left(\int−{xdx}\right)=\left(\frac{{x}^{\mathrm{2}} }{\mathrm{2}}\right){or}\left(\frac{−{x}^{\mathrm{2}} }{\mathrm{2}}\right)+\boldsymbol{{C}} \\…