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Category: Integration

advanced-calculus-prove-that-I-1-t-4-6t-2-1-ln-ln-t-1-t-2-3-dt-2G-pi-G-catalan-constant-

Question Number 125462 by mnjuly1970 last updated on 11/Dec/20 $$\:\:\:\:\:\:\:\:\:\:\:\:\:…\:\clubsuit{advanced}\:\:{calculus}\clubsuit… \\ $$$$\:\:\:\blacklozenge\blacklozenge\:{prove}\:{that}: \\ $$$$\:\:\:\:\:\mathrm{I}=\int_{\mathrm{1}} ^{\:\infty} \frac{\left({t}^{\mathrm{4}} −\mathrm{6}{t}^{\mathrm{2}} +\mathrm{1}\right){ln}\left({ln}\left({t}\right)\right)}{\left(\mathrm{1}+{t}^{\mathrm{2}} \right)^{\mathrm{3}} }{dt}=\frac{\mathrm{2G}}{\pi} \\ $$$$\:\:\mathrm{G}\::\:\:{catalan}\:\:{constant}… \\ $$ Answered…

advanced-calculus-evaluate-n-2-2n-2-n-

Question Number 125458 by mnjuly1970 last updated on 11/Dec/20 $$\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:…{advanced}\:\:{calculus}… \\ $$$$\:\:\:\:\:\:\:\:{evaluate}\:::: \\ $$$$\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\underset{{n}=\mathrm{2}} {\overset{\infty} {\sum}}\left\{\:\frac{\zeta\:\left(\mathrm{2}{n}\:\right)}{\mathrm{2}^{\:{n}} }\:\right\}\:=?? \\ $$$$ \\ $$ Answered by Dwaipayan Shikari…

Question-190987

Question Number 190987 by Rupesh123 last updated on 15/Apr/23 Answered by 07049753053 last updated on 16/Apr/23 $$\boldsymbol{\mathrm{let}}\:\boldsymbol{\mathrm{x}}^{\mathrm{2}} =\boldsymbol{\mathrm{u}}\:\boldsymbol{\mathrm{dx}}=\frac{\boldsymbol{\mathrm{du}}}{\mathrm{2}\sqrt{\boldsymbol{\mathrm{u}}}} \\ $$$$\frac{\mathrm{1}}{\mathrm{2}}\int_{\mathrm{0}} ^{\infty} \frac{\boldsymbol{\mathrm{e}}^{−\boldsymbol{\mathrm{u}}} \boldsymbol{\mathrm{sin}}\left(\boldsymbol{\mathrm{u}}\right)}{\boldsymbol{\mathrm{u}}\sqrt{\boldsymbol{\mathrm{u}}}}\boldsymbol{\mathrm{du}}=\frac{\mathrm{1}}{\mathrm{2}}\int_{\mathrm{0}} ^{\infty} \boldsymbol{\mathrm{u}}^{−\frac{\mathrm{3}}{\mathrm{2}}}…

2x-2-3x-3-x-1-x-2-2x-5-dx-

Question Number 125440 by bramlexs22 last updated on 11/Dec/20 $$\:\:\:\:\:\:\:\:\:\:\:\int\frac{\mathrm{2}{x}^{\mathrm{2}} −\mathrm{3}{x}−\mathrm{3}}{\left({x}−\mathrm{1}\right)\left({x}^{\mathrm{2}} −\mathrm{2}{x}+\mathrm{5}\right)}\:{dx}\: \\ $$ Answered by Ar Brandon last updated on 11/Dec/20 $$\mathrm{f}\left(\mathrm{x}\right)=\frac{\mathrm{2x}^{\mathrm{2}} −\mathrm{3x}−\mathrm{3}}{\left(\mathrm{x}−\mathrm{1}\right)\left(\mathrm{x}^{\mathrm{2}} −\mathrm{2x}+\mathrm{5}\right)}=\frac{\mathrm{a}}{\mathrm{x}−\mathrm{1}}+\frac{\mathrm{bx}+\mathrm{c}}{\mathrm{x}^{\mathrm{2}}…

2x-1-20-

Question Number 59905 by Sardor2211 last updated on 15/May/19 $$\int\left(\mathrm{2x}−\mathrm{1}\hat {\right)}\mathrm{20} \\ $$ Commented by maxmathsup by imad last updated on 16/May/19 $${if}\:{you}\:{mean}\:{I}\:=\int\:\left(\mathrm{2}{x}−\mathrm{1}\right)^{\mathrm{20}} {dx}\:\Rightarrow{I}\:=\int\:\sum_{{k}=\mathrm{0}} ^{\mathrm{20}}…