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Category: Integration

Show-that-sech-x-tanh-x-x-2-cosh-x-

Question Number 190937 by Spillover last updated on 14/Apr/23 $$\mathrm{Show}\:\:\mathrm{that}\:\: \\ $$$$\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\int\:\:\frac{\mathrm{sech}\:\sqrt{\mathrm{x}}\:\mathrm{tanh}\:\sqrt{\mathrm{x}}}{\:\sqrt{\mathrm{x}}}=−\frac{\mathrm{2}}{\mathrm{cosh}\:\sqrt{\mathrm{x}}} \\ $$ Answered by ARUNG_Brandon_MBU last updated on 15/Apr/23 $${I}=\int\frac{\left(\mathrm{sech}\sqrt{{x}}\right)\left(\mathrm{tanh}\sqrt{{x}}\right)}{\:\sqrt{{x}}}{dx}=\int\frac{\mathrm{sinh}\sqrt{{x}}}{\:\sqrt{{x}}\mathrm{cosh}^{\mathrm{2}} \sqrt{{x}}}{dx} \\ $$$${t}=\mathrm{cosh}\sqrt{{x}}\:\Rightarrow{dt}=\frac{\mathrm{sinh}\sqrt{{x}}}{\mathrm{2}\sqrt{{x}}}{dx}…

Question-59846

Question Number 59846 by bhanukumarb2@gmail.com last updated on 15/May/19 Commented by MJS last updated on 15/May/19 $$\mathrm{do}\:\mathrm{you}\:\mathrm{have}\:\mathrm{an}\:\mathrm{answer}? \\ $$$$\mathrm{it}\:\mathrm{seems}\:\mathrm{to}\:\mathrm{be}\:\frac{\mathrm{1}}{\mathrm{4}} \\ $$ Commented by bhanukumarb2@gmail.com last…

Question-59834

Question Number 59834 by bhanukumarb2@gmail.com last updated on 15/May/19 Answered by tanmay last updated on 15/May/19 $$\mathrm{1}>{sin}\left(\mathrm{3}{x}\right)>−\mathrm{1} \\ $$$$\:\left({x}−\mathrm{1}\right)\geqslant{x}+{sin}\left(\mathrm{3}{x}\right)\geqslant{x}−\mathrm{1} \\ $$$$\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\int_{\mathrm{0}} ^{\pi} {sin}^{\mathrm{4}} \left({x}+\mathrm{1}\right){dx}\geqslant\int_{\mathrm{0}} ^{\pi}…

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

Question Number 59807 by aliesam last updated on 15/May/19 $$\int\frac{{x}−\mathrm{1}}{\:\sqrt{\mathrm{2}{x}−{x}^{\mathrm{2}} }}\:{dx} \\ $$ Answered by tanmay last updated on 15/May/19 $${t}^{\mathrm{2}} =\mathrm{2}{x}−{x}^{\mathrm{2}} \\ $$$$\mathrm{2}{tdt}=\left(\mathrm{2}−\mathrm{2}{x}\right){dx} \\…

Question-59803

Question Number 59803 by bhanukumarb2@gmail.com last updated on 15/May/19 Commented by maxmathsup by imad last updated on 16/May/19 $${let}\:{A}\:=\frac{\mathrm{1}}{\pi^{\mathrm{2}} }\:\int_{\mathrm{0}} ^{\infty} \:\frac{\left({lnx}\right)^{\mathrm{2}} }{\:\sqrt{{x}}\left(\mathrm{1}−{x}\right)^{\mathrm{2}} }\:{dx}\:\Rightarrow\pi^{\mathrm{2}} \:{A}\:=_{\sqrt{{x}}={t}}…