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Category: Integration

Advanced-Calculus-Calculate-i-I-0-1-ln-x-ln-1-x-dx-ii-J-0-1-Li-2-1-x-2-Note-Li-2-x-

Question Number 147287 by mnjuly1970 last updated on 19/Jul/21 $$ \\ $$$$\:\:\:\:\:\:\:\:\:\:\:\:\:…\mathrm{Advanced}\:\:\mathrm{Calculus}… \\ $$$$ \\ $$$$\:\:\:\:\:\:\mathrm{C}{alculate}\:::\:\:\:\:\begin{cases}{\:\:\mathrm{i}\:::\:\:\:\:\:\:\mathrm{I}\::=\:\int_{\mathrm{0}} ^{\:\mathrm{1}} \mathrm{ln}\left(\mathrm{x}\right).\mathrm{ln}\left(\mathrm{1}+\mathrm{x}\right)\:\mathrm{dx}}\\{\:\:\mathrm{ii}\:::\:\:\:\:\:\mathrm{J}\::=\:\int_{\mathrm{0}} ^{\:\mathrm{1}} \mathrm{Li}_{\:\mathrm{2}} \left(\:\mathrm{1}−\:\mathrm{x}^{\:\mathrm{2}} \right)\:=?}\end{cases} \\ $$$$\:\:\:\:\:\:\:\:\:\:\mathrm{Note}::\:\:\:\mathrm{Li}_{\mathrm{2}} \:\left(\mathrm{x}\right)\:=\:\underset{{n}=\mathrm{1}}…

dx-cos-3-x-sin-3-x-

Question Number 81739 by TANMAY PANACEA last updated on 15/Feb/20 $$\int\frac{{dx}}{{cos}^{\mathrm{3}} {x}−{sin}^{\mathrm{3}} {x}} \\ $$ Commented by abdomathmax last updated on 16/Feb/20 $${I}\:=\int\:\:\frac{{dx}}{{cos}^{\mathrm{3}} {x}−{sin}^{\mathrm{3}} {x}}\:=\int\:\frac{{dx}}{\left({cosx}−{sinx}\right)\left({cos}^{\mathrm{2}}…

1-find-dx-x-2-5-x-3-9-2-calculate-4-dx-x-2-5-x-3-9-

Question Number 81719 by mathmax by abdo last updated on 14/Feb/20 $$\left.\mathrm{1}\right)\:{find}\:\int\:\:\:\frac{{dx}}{\left({x}+\mathrm{2}\right)^{\mathrm{5}} \left({x}−\mathrm{3}\right)^{\mathrm{9}} } \\ $$$$ \\ $$$$\left.\mathrm{2}\right)\:{calculate}\:\int_{\mathrm{4}} ^{+\infty} \:\frac{{dx}}{\left({x}+\mathrm{2}\right)^{\mathrm{5}} \left({x}−\mathrm{3}\right)^{\mathrm{9}} } \\ $$ Commented…

find-U-n-0-e-nx-2-x-2-n-2-dx-n-1-nature-of-U-n-and-nU-n-

Question Number 147203 by mathmax by abdo last updated on 18/Jul/21 $$\mathrm{find}\:\mathrm{U}_{\mathrm{n}} =\int_{\mathrm{0}} ^{\infty} \:\frac{\mathrm{e}^{−\mathrm{nx}^{\mathrm{2}} } }{\mathrm{x}^{\mathrm{2}} \:+\mathrm{n}^{\mathrm{2}} }\mathrm{dx}\:\:\:\:\:\left(\mathrm{n}\geqslant\mathrm{1}\right) \\ $$$$\mathrm{nature}\:\mathrm{of}\:\Sigma\mathrm{U}_{\mathrm{n}} \:\mathrm{and}\:\Sigma\:\mathrm{nU}_{\mathrm{n}} \\ $$ Answered…