Question Number 95198 by Fikret last updated on 23/May/20 $$\int\sqrt{\frac{{x}+\mathrm{2}}{{x}−\mathrm{3}}}{dx}=? \\ $$ Answered by MJS last updated on 23/May/20 $$\int\sqrt{\frac{{x}+\mathrm{2}}{{x}−\mathrm{3}}}{dx}= \\ $$$$\:\:\:\:\:\left[{t}=\sqrt{\frac{{x}+\mathrm{2}}{{x}−\mathrm{3}}}\:\rightarrow\:{dx}=−\frac{\mathrm{2}}{\mathrm{5}}\sqrt{\left({x}−\mathrm{3}\right)^{\mathrm{3}} \left({x}+\mathrm{2}\right)}\right] \\ $$$$=−\mathrm{10}\int\frac{{t}^{\mathrm{2}}…
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Question Number 160734 by mnjuly1970 last updated on 05/Dec/21 $$ \\ $$$$\:\:\:\:\:\:\:\:\:\:\:\:#\:\mathrm{Advanced}\:\:\:\mathrm{Calculus}\:# \\ $$$$\:\:\:\:\:\:\:\:\Phi\:=\:\int_{\mathrm{0}} ^{\:\mathrm{1}} \left(\frac{{ln}^{\:} \:\left(\:\frac{\mathrm{1}}{\mathrm{1}−\:{x}}\:\:\right)}{{x}}\:\right)^{\:\mathrm{3}} {dx}\:\overset{?} {=}\:\mathrm{3}\:\left(\:\zeta\:\left(\mathrm{2}\:\right)\:+\:\zeta\:\left(\mathrm{3}\:\right)\right) \\ $$$$\:\:\:\:\:\:−−−−\:\:{solution}−−−− \\ $$$$\:\:\:\:\:\:\:\:\Phi\:\overset{\mathrm{I}.\mathrm{B}.\mathrm{P}} {=}\:\left[\:\frac{\:\mathrm{1}}{\mathrm{2}{x}^{\:\mathrm{2}} }\:{ln}^{\:\mathrm{3}}…
Question Number 95199 by Fikret last updated on 23/May/20 $$\int_{−\pi} ^{\pi} {cosxln}\frac{\mathrm{1}+{x}}{\mathrm{1}−{x}}\:{dx}=? \\ $$ Answered by mr W last updated on 23/May/20 $${f}\left({x}\right)=\mathrm{cos}\:{x}\:\mathrm{ln}\:\frac{\mathrm{1}+{x}}{\mathrm{1}−{x}} \\ $$$${f}\left(−{x}\right)=\mathrm{cos}\:\left(−{x}\right)\:\mathrm{ln}\:\frac{\mathrm{1}+\left(−{x}\right)}{\mathrm{1}−\left(−{x}\right)}…
Question Number 95190 by mhmd last updated on 23/May/20 Commented by mhmd last updated on 23/May/20 $${find}\:{S}_{\mathrm{2}} +{S}_{\mathrm{4}} \\ $$$${help}\:{me}\:{sir} \\ $$ Answered by prakash…
Question Number 95115 by bobhans last updated on 23/May/20 $$\int\:\mathrm{e}^{\:{x}} \:\sqrt{\mathrm{1}+{e}^{\:\mathrm{2}{x}} }\:{dx}\:=\:?\: \\ $$ Answered by i jagooll last updated on 23/May/20 Commented by bobhans…
Question Number 160645 by tounghoungko last updated on 03/Dec/21 $$\:\int\:\mathrm{sin}\:^{\mathrm{8}} \mathrm{x}\:\mathrm{dx}\:=? \\ $$ Commented by swamy1234 last updated on 04/Dec/21 good Answered by puissant last…
Question Number 29574 by yesasitya22@gmail.com last updated on 10/Feb/18 $$\int{x}^{\mathrm{6}} −\mathrm{1}/{x}^{\mathrm{2}} −\mathrm{1}{dx} \\ $$ Commented by tawa tawa last updated on 10/Feb/18 $$\int\:\frac{\mathrm{x}^{\mathrm{6}} \:−\:\mathrm{1}}{\mathrm{x}^{\mathrm{2}} \:−\:\mathrm{1}}\:\mathrm{dx}…
Question Number 160636 by mnjuly1970 last updated on 03/Dec/21 $$ \\ $$$$\:\:\:\:\:\:\:\:{prove}\:{that} \\ $$$$ \\ $$$$\:\:\:\:\:\:\:\:\int_{\mathrm{0}\:\:} ^{\:\mathrm{1}} \left(\frac{{tanh}^{\:−\mathrm{1}} \left(\:{x}\:\right)}{{x}}\:\right)^{\:\mathrm{2}} =\:\zeta\:\left(\:\mathrm{2}\:\right)\:\:\:\:\:\:\:\:\blacksquare\:{m}.{n}\:\:\:\: \\ $$$$ \\ $$$$ \\…
Question Number 29553 by abdo imad last updated on 09/Feb/18 $${let}\:{put}\:\:{I}\left({x}\right)=\:\int_{{x}} ^{+\infty} \:\frac{{sin}^{\mathrm{3}} {t}}{{t}^{\mathrm{2}} }{dt}\:\:{with}\:{x}>\mathrm{0} \\ $$$${find}\:{lim}_{{x}\rightarrow\mathrm{0}^{+} } \:\:{I}\left({x}\right) \\ $$$$\left.\mathrm{2}\right)\:{find}\:\int_{\mathrm{0}} ^{\infty} \:\:\frac{{sin}^{\mathrm{3}} {t}}{{t}^{\mathrm{2}} }{dt}\:.…