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Category: Integration

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}}…

let-f-a-dx-a-2-x-2-3-with-a-gt-0-1-calculate-f-a-2-calculste-also-g-a-dx-a-2-x-2-4-3-find-the-values-of-integrals-0-dx-x-2-1-3

Question Number 63782 by mathmax by abdo last updated on 09/Jul/19 $${let}\:{f}\left({a}\right)\:=\int_{−\infty} ^{+\infty} \:\:\:\frac{{dx}}{\left({a}^{\mathrm{2}} +{x}^{\mathrm{2}} \right)^{\mathrm{3}} }\:\:\:{with}\:{a}>\mathrm{0} \\ $$$$\left.\mathrm{1}\right)\:{calculate}\:{f}\left({a}\right) \\ $$$$\left.\mathrm{2}\right){calculste}\:{also}\:{g}\left({a}\right)\:=\int_{−\infty} ^{+\infty} \:\:\:\frac{{dx}}{\left({a}^{\mathrm{2}} \:+{x}^{\mathrm{2}} \right)^{\mathrm{4}}…