Question Number 57324 by turbo msup by abdo last updated on 02/Apr/19 $${we}\:{want}\:{to}\:{find}\:{the}\:{vslue}\:{of} \\ $$$${I}\:=\int_{\mathrm{0}} ^{\mathrm{1}} \:\frac{{ln}\left(\mathrm{1}+{x}\right)}{\mathrm{1}+{x}^{\mathrm{2}} }\:{dx}\:{let} \\ $$$${A}=\int\int_{{W}} \frac{{x}}{\left(\mathrm{1}+{x}^{\mathrm{2}} \right)\left(\mathrm{1}+{xy}\right)}{dxdy} \\ $$$${with}\:{W}=\left[\mathrm{0},\mathrm{1}\right]^{\mathrm{2}} \\…
Question Number 57323 by turbo msup by abdo last updated on 02/Apr/19 $${calculate}\:\int\int_{{D}} \:\:\frac{{x}+{y}}{\mathrm{3}+\sqrt{{x}^{\mathrm{2}} \:+{y}^{\mathrm{2}} }}{dxdy} \\ $$$${with}\:{D}=\left\{\left({x},{y}\right)\in{R}^{\mathrm{2}} /{x}^{\mathrm{2}} \:+{y}^{\mathrm{2}} \leqslant\mathrm{2}\right. \\ $$$$\left.{and}\:{x}\geqslant\mathrm{0}\:,{y}\geqslant\mathrm{0}\right\} \\ $$…
Question Number 57321 by turbo msup by abdo last updated on 02/Apr/19 $${calculate}\:\int\int_{{D}} \left({x}−{y}\right)\sqrt{{x}^{\mathrm{2}} \:+{y}^{\mathrm{2}} }{dxdy} \\ $$$${with}\:{D}\:=\left\{\:\left({x},{y}\right)\in{R}^{\mathrm{2}} /{x}^{\mathrm{2}} \:+{y}^{\mathrm{2}} \:\leqslant\mathrm{2}\:{and}\:{x}\geqslant\mathrm{0}\right\} \\ $$ Commented by…
Question Number 57320 by turbo msup by abdo last updated on 02/Apr/19 $${calculate}\:\int\int_{{D}} {xy}\:{e}^{−{x}^{\mathrm{2}} −{y}^{\mathrm{2}} } \:{dxdy} \\ $$$${with}\:{D}=\left\{\left({x},{y}\right)\in{R}^{\mathrm{2}} /\:\mathrm{0}\leqslant{x}\leqslant\mathrm{2}\:{and}\right. \\ $$$$\left.\mathrm{1}\leqslant{y}\leqslant\mathrm{3}\right\} \\ $$ Commented…
Question Number 122853 by rs4089 last updated on 20/Nov/20 Commented by Dwaipayan Shikari last updated on 20/Nov/20 $${y}\:\:\:\bigtriangleup{y}\:\bigtriangleup^{\mathrm{2}} {y} \\ $$$$\mathrm{2} \\ $$$$\:\:\:\:\:\:\mathrm{3} \\ $$$$\mathrm{5}\:\:\:\:\:\:\:\:\:\:\:\:\:\:\mathrm{2}…
Question Number 57319 by turbo msup by abdo last updated on 02/Apr/19 $${calculate}\:\int\int_{{D}} \:{e}^{{x}−{y}} \:{dxdy} \\ $$$${with}\:{D}=\left\{\left({x},{y}\right)\in{R}^{\mathrm{2}} \:/\mid{x}\mid<\mathrm{1}\:{and}\:\mathrm{0}\leqslant{y}\leqslant\mathrm{1}\right\} \\ $$ Commented by maxmathsup by imad…
Question Number 188384 by universe last updated on 28/Feb/23 $$\:\:\: \\ $$$$\:\:\:\mathrm{if}\:\:\:\:\int_{\mathrm{0}} ^{\infty} {e}^{−{ax}} {dx}\:\:=\:\:\frac{\mathrm{1}}{{a}} \\ $$$$\:\:\:\:\:\:\:\:{show}\:{that}\:\:\int_{\mathrm{0}} ^{\infty} {e}^{−{ax}} \:{x}^{{n}} {dx}\:\:=\:\:\frac{{n}!}{{a}^{{n}+\mathrm{1}} } \\ $$ Answered…
Question Number 188381 by mnjuly1970 last updated on 28/Feb/23 $$ \\ $$$$\:\:\:\:\:\:\:\:\:\:\mathrm{Advanced}\:\:\mathrm{calculus} \\ $$$$ \\ $$$$\:\:\:\:\:\mathrm{Find}\:\:\mathrm{the}\:\:\mathrm{value}\:\:\mathrm{of}\:\mathrm{the}\:\mathrm{following}\:\mathrm{series}. \\ $$$$ \\ $$$$\:\:\:\:\:\:\:\:\:\Omega\:=\:\underset{{n}=\mathrm{1}} {\overset{\infty} {\sum}}\:\frac{\:\left(−\mathrm{1}\right)^{\:{n}} \:\zeta\:\left(\:{n}\:\right)}{{n}.\:\mathrm{2}^{\:{n}} }\:=\:? \\…
Question Number 188380 by Shlock last updated on 28/Feb/23 Answered by SEKRET last updated on 28/Feb/23 $$\:\:\boldsymbol{\mathrm{f}}\left(\boldsymbol{\mathrm{t}}\right)=\int_{\mathrm{0}} ^{\:\boldsymbol{\pi}} \frac{\boldsymbol{\mathrm{ln}}\left(\mathrm{1}+\boldsymbol{\mathrm{t}}\centerdot\boldsymbol{\mathrm{sin}}\left(\boldsymbol{\mathrm{a}}\right)\centerdot\boldsymbol{\mathrm{cos}}\left(\boldsymbol{\mathrm{x}}\right)\right)}{\boldsymbol{\mathrm{cos}}\left(\boldsymbol{\mathrm{x}}\right)}\boldsymbol{\mathrm{dx}} \\ $$$$\:\boldsymbol{\mathrm{f}}\:'\:\left(\boldsymbol{\mathrm{t}}\right)=\:\int_{\mathrm{0}} ^{\:\boldsymbol{\pi}} \:\frac{\boldsymbol{\mathrm{sin}}\left(\boldsymbol{\mathrm{a}}\right)\centerdot\boldsymbol{\mathrm{cos}}\left(\boldsymbol{\mathrm{x}}\right)}{\boldsymbol{\mathrm{cos}}\left(\boldsymbol{\mathrm{x}}\right)\centerdot\left(\mathrm{1}+\boldsymbol{\mathrm{t}}\centerdot\boldsymbol{\mathrm{sin}}\left(\boldsymbol{\mathrm{a}}\right)\centerdot\boldsymbol{\mathrm{cos}}\left(\boldsymbol{\mathrm{x}}\right)\right)}\boldsymbol{\mathrm{dx}} \\ $$$$\:\boldsymbol{\mathrm{f}}\:'\:\left(\boldsymbol{\mathrm{t}}\right)=\:\boldsymbol{\mathrm{sin}}\left(\boldsymbol{\mathrm{a}}\right)\centerdot\int_{\mathrm{0}}…
Question Number 122838 by bemath last updated on 20/Nov/20 $$\:\:\int\:\frac{{dx}}{\:\sqrt[{\mathrm{4}}]{\left({x}−\mathrm{1}\right)^{\mathrm{3}} \left({x}+\mathrm{2}\right)^{\mathrm{5}} }}\:? \\ $$ Answered by liberty last updated on 20/Nov/20 $$\:{because}\:\sqrt[{\mathrm{4}}]{\left({x}−\mathrm{1}\right)^{\mathrm{3}} \left({x}+\mathrm{2}\right)^{\mathrm{5}} }\:=\:\left({x}−\mathrm{1}\right)\left({x}+\mathrm{2}\right)\:\sqrt[{\mathrm{4}}]{\frac{{x}+\mathrm{2}}{{x}−\mathrm{1}}} \\…