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Category: Integration

prove-that-0-1-t-2p-1-ln-t-t-2-1-dt-pi-2-24-1-4-k-1-p-1-k-2-

Question Number 40886 by prof Abdo imad last updated on 28/Jul/18 $${prove}\:{that}\: \\ $$$$\int_{\mathrm{0}} ^{\mathrm{1}} \:\:\frac{{t}^{\mathrm{2}{p}+\mathrm{1}} {ln}\left({t}\right)}{{t}^{\mathrm{2}} −\mathrm{1}}{dt}\:=\frac{\pi^{\mathrm{2}} }{\mathrm{24}}\:−\frac{\mathrm{1}}{\mathrm{4}}\sum_{{k}=\mathrm{1}} ^{{p}} \:\frac{\mathrm{1}}{{k}^{\mathrm{2}} } \\ $$ Terms…

e-2x-2-x-dx-

Question Number 106391 by Ar Brandon last updated on 05/Aug/20 $$\int\mathrm{e}^{\mathrm{2x}^{\mathrm{2}} +\mathrm{x}} \mathrm{dx} \\ $$ Answered by Sarah85 last updated on 05/Aug/20 $${t}=\sqrt{\mathrm{2}}\left({x}+\frac{\mathrm{1}}{\mathrm{4}}\right)\:\Leftrightarrow\:{x}=\frac{\sqrt{\mathrm{2}}}{\mathrm{2}}{t}−\frac{\mathrm{1}}{\mathrm{4}}\:\Rightarrow\:{dx}=\frac{\sqrt{\mathrm{2}}}{\mathrm{2}}{dt} \\ $$$$\frac{\sqrt{\mathrm{2}}}{\mathrm{2}}\int\mathrm{e}^{{t}^{\mathrm{2}}…

dx-9-4x-2-using-the-trigonometric-substitution-

Question Number 171910 by Tawa11 last updated on 21/Jun/22 $$\int\:\frac{\mathrm{dx}}{\mathrm{9}\:\:\:−\:\:\:\mathrm{4x}^{\mathrm{2}} } \\ $$$$\mathrm{using}\:\mathrm{the}\:\mathrm{trigonometric}\:\mathrm{substitution}. \\ $$ Answered by aleks041103 last updated on 21/Jun/22 $$\int\frac{{dx}}{\mathrm{9}−\mathrm{4}{x}^{\mathrm{2}} }=\frac{\mathrm{1}}{\mathrm{9}}\int\frac{{dx}}{\mathrm{1}−\left(\frac{\mathrm{2}{x}}{\mathrm{3}}\right)^{\mathrm{2}} }=\frac{\mathrm{1}}{\mathrm{9}}\:\frac{\mathrm{3}}{\mathrm{2}}\int\frac{{du}}{\mathrm{1}−{u}^{\mathrm{2}}…