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Category: Integration

Evaluate-r-0-x-r-x-dx-

Question Number 27400 by Tinkutara last updated on 06/Jan/18 $${Evaluate}\:\underset{{r}} {\overset{\mathrm{0}} {\int}}\sqrt{\frac{{x}}{{r}−{x}}}\:{dx} \\ $$ Commented by Tinkutara last updated on 07/Jan/18 $${But}\:{this}\:{integral}\:{was}\:{used}\:{in}\:{a}\:{Physics} \\ $$$${question}\:{where}\:{its}\:{value}\:{can}'{t}\:{be} \\…

learning-distancing-ln-1-x-1-x-dx-

Question Number 92889 by i jagooll last updated on 09/May/20 $$\mathrm{learning}\:\mathrm{distancing} \\ $$$$\int\:\mathrm{ln}\left(\sqrt{\mathrm{1}+\mathrm{x}}+\sqrt{\mathrm{1}−\mathrm{x}}\right)\:\mathrm{dx} \\ $$ Commented by john santu last updated on 09/May/20 $$\mathrm{u}\:=\:\mathrm{ln}\left(\sqrt{\mathrm{1}+\mathrm{x}}\:+\sqrt{\mathrm{1}−\mathrm{x}}\right) \\…

prove-that-0-t-x-1-e-t-1-dt-x-x-with-x-n-1-1-n-x-and-x-0-t-x-1-e-t-dt-x-gt-1-

Question Number 27345 by abdo imad last updated on 05/Jan/18 $${prove}\:{that}\:\:\int_{\mathrm{0}} ^{\infty} \:\:\frac{{t}^{{x}−\mathrm{1}} }{{e}^{{t}} −\mathrm{1}}{dt}\:\:=\xi\left({x}\right)\Gamma\left({x}\right) \\ $$$${with}\:\xi\left({x}\right)=\:\sum_{{n}=\mathrm{1}} ^{\propto} \:\frac{\mathrm{1}}{{n}^{{x}} }\:\:\:{and}\:\Gamma\left({x}\right)=\int_{\mathrm{0}} ^{\infty} \:{t}^{{x}−\mathrm{1}} \:{e}^{−{t}} {dt} \\…