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Category: Integration

calculate-A-0-x-2017-1-x-2019-dx-and-B-0-x-2019-1-x-2021-dx-calculate-the-fraction-A-B-

Question Number 63892 by mathmax by abdo last updated on 10/Jul/19 $${calculate}\:{A}=\int_{\mathrm{0}} ^{\infty} \:\:\frac{{x}^{\mathrm{2017}} }{\mathrm{1}+{x}^{\mathrm{2019}} }\:{dx}\:\:{and}\:{B}\:=\int_{\mathrm{0}} ^{\infty} \:\frac{{x}^{\mathrm{2019}} }{\mathrm{1}+{x}^{\mathrm{2021}} }\:{dx} \\ $$$${calculate}\:{the}\:{fraction}\:\frac{{A}}{{B}} \\ $$ Commented…

ln-x-ln-1-x-ln-1-2x-dx-

Question Number 63883 by mmkkmm000m last updated on 10/Jul/19 $$\int{ln}\left({x}\right){ln}\left(\mathrm{1}−{x}\right){ln}\left(\mathrm{1}−\mathrm{2}{x}\right){dx} \\ $$ Commented by mathmax by abdo last updated on 12/Jul/19 $${let}\:{A}\:=\int\:{ln}\left({x}\right){ln}\left(\mathrm{1}−{x}\right){ln}\left(\mathrm{1}−\mathrm{2}{x}\right)\:{dx}\:\:{we}\:{have} \\ $$$${ln}^{'} \left(\mathrm{1}−{u}\right)\:=−\frac{\mathrm{1}}{\mathrm{1}−{u}}\:=−\sum_{{n}=\mathrm{0}}…

prove-that-0-1-arctan-x-cot-pix-2-dx-3-ln-2-2-2pi-lnpi-ln2-pi-0-ln-1-x-2-e-2pix-1-dx-

Question Number 63852 by aliesam last updated on 10/Jul/19 $${prove}\:{that} \\ $$$$ \\ $$$$\int_{\mathrm{0}} ^{\mathrm{1}} {arctan}\left({x}\right)\:{cot}\left(\frac{\pi{x}}{\mathrm{2}}\right)\:{dx}\:=\:\frac{\mathrm{3}\:{ln}^{\mathrm{2}} \left(\mathrm{2}\right)}{\mathrm{2}\pi}+\frac{{ln}\pi\:{ln}\mathrm{2}}{\pi}+\int_{\mathrm{0}} ^{\infty} \frac{{ln}\left(\mathrm{1}+{x}^{\mathrm{2}} \right)}{{e}^{\mathrm{2}\pi{x}} +\mathrm{1}}\:{dx} \\ $$ Terms of…

x-x-1-dx-

Question Number 129377 by pipin last updated on 15/Jan/21 $$\int\frac{\:\sqrt{\boldsymbol{\mathrm{x}}}}{\:\sqrt{\boldsymbol{\mathrm{x}}-\mathrm{1}}}\mathrm{dx}\:=\:… \\ $$ Answered by Ar Brandon last updated on 15/Jan/21 $$\mathcal{I}=\int\frac{\sqrt{\mathrm{x}}}{\:\sqrt{\mathrm{x}−\mathrm{1}}}\mathrm{dx}\:,\:\mathrm{x}=\mathrm{t}^{\mathrm{2}} \:\Rightarrow\mathrm{dx}=\mathrm{2tdt} \\ $$$$\:\:\:=\mathrm{2}\int\frac{\mathrm{t}^{\mathrm{2}} }{\:\sqrt{\mathrm{t}^{\mathrm{2}}…