Question Number 52948 by gunawan last updated on 15/Jan/19 $$\mathrm{If}\:\:\:{f}\left({x}\right)\:=\underset{\:\mathrm{1}} {\overset{{x}} {\int}}\:\frac{\mathrm{log}\:{t}}{\mathrm{1}+{t}}\:{dt},\:\mathrm{then} \\ $$$$\:{f}\left({x}\right)+{f}\:\left(\frac{\mathrm{1}}{{x}}\:\right)=\frac{\mathrm{1}}{\mathrm{2}}\left(\mathrm{log}\:{x}\right)^{\mathrm{2}} \\ $$ Commented by maxmathsup by imad last updated on 15/Jan/19…
Question Number 52947 by gunawan last updated on 15/Jan/19 $$\underset{\:\mathrm{0}} {\overset{\pi} {\int}}\:\:\frac{{x}\:\mathrm{tan}\:{x}}{\mathrm{sec}\:{x}+\mathrm{cos}\:{x}}\:{dx}\:= \\ $$ Answered by tanmay.chaudhury50@gmail.com last updated on 15/Jan/19 $$\int_{\mathrm{0}} ^{\pi} \frac{{x}×\frac{{sinx}}{{cosx}}}{\frac{\mathrm{1}}{{cosx}}+{cosx}}{dx} \\…
Question Number 52946 by gunawan last updated on 15/Jan/19 $$\underset{\:\mathrm{0}} {\overset{\pi/\mathrm{2}} {\int}}\:\:\mathrm{log}\:\mathrm{sin}\:\mathrm{2}{x}\:{dx}\:= \\ $$ Commented by maxmathsup by imad last updated on 15/Jan/19 $${let}\:{A}\:=\int_{\mathrm{0}} ^{\frac{\pi}{\mathrm{2}}}…
Question Number 52600 by Tip Top last updated on 10/Jan/19 $$\mathrm{If}\:{P}_{{n}} \:\mathrm{denotes}\:\mathrm{the}\:\mathrm{product}\:\mathrm{of}\:\mathrm{the}\:\mathrm{binomial}\: \\ $$$$\mathrm{coefficients}\:\mathrm{in}\:\mathrm{the}\:\mathrm{expansion}\:\mathrm{of}\:\left(\mathrm{1}+{x}\right)^{{n}} , \\ $$$$\mathrm{then}\:\frac{{P}_{{n}+\mathrm{1}} }{{P}_{{n}} }\:\mathrm{equals} \\ $$ Commented by Abdo msup.…
Question Number 51729 by shivshant0409 last updated on 30/Dec/18 $$\mathrm{81}^{\frac{\mathrm{1}}{\mathrm{4}}} ×\:\mathrm{9}^{\frac{\mathrm{3}}{\mathrm{2}}} ×\:\mathrm{27}^{\frac{\mathrm{4}}{\mathrm{3}}} =\:\_\_\_\_\_ \\ $$ Answered by aseerimad last updated on 30/Dec/18 $$\mathrm{81}^{\frac{\mathrm{1}}{\mathrm{4}}} ×\mathrm{9}^{\frac{\mathrm{3}}{\mathrm{2}}} ×\mathrm{27}^{\frac{\mathrm{4}}{\mathrm{3}}}…
Question Number 116006 by DELETED last updated on 30/Sep/20 $$\mathrm{3}\left(\mathrm{sin}\:{x}−\mathrm{cos}\:{x}\right)^{\mathrm{4}} +\mathrm{6}\left(\mathrm{sin}\:{x}+\mathrm{cos}\:{x}\right)^{\mathrm{2}} \\ $$$$\:\:\:\:\:\:\:\:\:\:\:\:+\mathrm{4}\left(\mathrm{sin}^{\mathrm{6}} {x}+\mathrm{cos}^{\mathrm{6}} {x}\right)\:=\:\_\_\_\_\_. \\ $$ Answered by mindispower last updated on 30/Sep/20 $$\left({sin}^{\mathrm{2}}…
Question Number 115695 by ZiYangLee last updated on 27/Sep/20 $$\mathrm{The}\:\mathrm{product}\:\mathrm{of}\:\mathrm{all}\:\mathrm{the}\:\mathrm{solutions}\:\mathrm{of}\:\mathrm{the} \\ $$$$\mathrm{equation}\:\:\mid\left({x}−\mathrm{2}\right)^{\mathrm{2}} \mid−\mathrm{3}\mid{x}−\mathrm{2}\mid+\mathrm{2}=\mathrm{0}\:\:\mathrm{is} \\ $$ Answered by MJS_new last updated on 27/Sep/20 $${x}=\mathrm{0}\:\mathrm{is}\:\mathrm{a}\:\mathrm{solution}\:\Rightarrow\:\mathrm{answer}\:\mathrm{is}\:\mathrm{0} \\ $$…
Question Number 115685 by ZiYangLee last updated on 27/Sep/20 $$\mathrm{If}\:\:\alpha,\:\beta,\:\gamma,\:\delta\:\:\mathrm{are}\:\mathrm{the}\:\mathrm{smallest}\:\mathrm{positive}\: \\ $$$$\mathrm{angles}\:\mathrm{in}\:\mathrm{ascending}\:\mathrm{order}\:\mathrm{of} \\ $$$$\mathrm{magnitude}\:\mathrm{which}\:\mathrm{have}\:\mathrm{their}\:\mathrm{sines}\: \\ $$$$\mathrm{equal}\:\mathrm{to}\:\mathrm{the}\:\mathrm{positive}\:\mathrm{quantity}\:{k},\:\mathrm{then} \\ $$$$\mathrm{the}\:\mathrm{value}\:\mathrm{of}\: \\ $$$$\mathrm{4}\:\mathrm{sin}\:\frac{\alpha}{\mathrm{2}}+\mathrm{3}\:\mathrm{sin}\:\frac{\beta}{\mathrm{2}}+\mathrm{2}\:\mathrm{sin}\:\frac{\gamma}{\mathrm{2}}+\mathrm{sin}\:\frac{\delta}{\mathrm{2}}\:\mathrm{is} \\ $$$$\mathrm{equal}\:\mathrm{to} \\ $$ Answered…
Question Number 50132 by CIRCLE001 last updated on 14/Dec/18 $$\mathrm{Let}\:{f}\:\:\mathrm{be}\:\mathrm{a}\:\mathrm{positive}\:\mathrm{function}.\:\mathrm{Let} \\ $$$${I}_{\mathrm{1}} =\underset{\mathrm{1}−{k}} {\overset{{k}} {\int}}{x}\:{f}\left\{{x}\left(\mathrm{1}−{x}\right\}\:{dx},\:\right. \\ $$$${I}_{\mathrm{2}} =\underset{\mathrm{1}−{k}} {\overset{{k}} {\int}}{x}\:{f}\left\{{x}\left(\mathrm{1}−{x}\right\}\:{dx},\:\right. \\ $$$$\mathrm{where}\:\mathrm{2}{k}−\mathrm{1}>\mathrm{0}.\:\mathrm{Then}\:\frac{{I}_{\mathrm{1}} }{{I}_{\mathrm{2}} }\:\:\mathrm{is} \\…
Question Number 115404 by EvoneAkashi last updated on 25/Sep/20 $$\:\underset{\:\mathrm{0}} {\overset{\infty} {\int}}\:\:\frac{\mathrm{1}}{\mathrm{1}+{x}^{\mathrm{4}} }\:{dx}\:= \\ $$ Answered by mathmax by abdo last updated on 25/Sep/20 $$\mathrm{let}\:\mathrm{I}\:=\int_{\mathrm{0}}…