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Category: Integration

Prove-that-e-e-2-lnx-1-lnx-2-dx-e-6-2e-3-

Question Number 17475 by Arnab Maiti last updated on 06/Jul/17 $$\mathrm{Prove}\:\mathrm{that}\:\int_{\mathrm{e}} ^{\mathrm{e}^{\mathrm{2}} } \frac{\mathrm{ln}{x}}{\left(\mathrm{1}+\mathrm{ln}{x}\right)^{\mathrm{2}} }\mathrm{d}{x}=\frac{\mathrm{e}}{\mathrm{6}}\left(\mathrm{2e}−\mathrm{3}\right) \\ $$ Answered by ajfour last updated on 06/Jul/17 $$\mathrm{lnx}=\mathrm{t}\:\:\:\Rightarrow\:\:\:\frac{\mathrm{dx}}{\mathrm{x}}=\mathrm{dt}\:\:\mathrm{or}\:\:\mathrm{dx}=\mathrm{e}^{\mathrm{t}}…

Prove-that-6-11-dx-x-2-x-3-2ln-3-2-2-2-3-

Question Number 17472 by Arnab Maiti last updated on 06/Jul/17 $$\mathrm{Prove}\:\mathrm{that}\:\int_{\mathrm{6}} ^{\mathrm{11}} \frac{\mathrm{dx}}{\:\sqrt{\left(\mathrm{x}−\mathrm{2}\right)\left(\mathrm{x}−\mathrm{3}\right)}}=\mathrm{2ln}\frac{\mathrm{3}+\mathrm{2}\sqrt{\mathrm{2}}}{\mathrm{2}+\sqrt{\mathrm{3}}} \\ $$ Answered by sma3l2996 last updated on 06/Jul/17 $${I}=\int_{\mathrm{6}} ^{\mathrm{11}} \frac{{dx}}{\:\sqrt{{x}^{\mathrm{2}}…