Question Number 55784 by gunawan last updated on 04/Mar/19 $$\mathrm{If}\:{n}\:\mathrm{an}\:\mathrm{odd}\:\mathrm{natural}\:\mathrm{number},\:\mathrm{then} \\ $$$$\underset{{r}=\mathrm{0}} {\overset{{n}} {\sum}}\:\frac{\left(−\mathrm{1}\right)^{{r}} }{\:^{{n}} {C}_{{r}} \:}\:\mathrm{equals} \\ $$ Answered by tanmay.chaudhury50@gmail.com last updated on…
Question Number 55780 by gunawan last updated on 04/Mar/19 $$\int\:\frac{\mathrm{1}}{\:\sqrt{{x}^{\mathrm{2}} +\mathrm{2}{x}+\mathrm{1}}}\:{dx}=\:{A}\:\mathrm{log}\:\mid{x}+\mathrm{1}\mid+{C}\:\mathrm{for} \\ $$$${x}>−\mathrm{1},\:\mathrm{then}\:{A}=\_\_\_\_\_. \\ $$ Commented by maxmathsup by imad last updated on 04/Mar/19 $$\int\:\frac{{dx}}{\:\sqrt{{x}^{\mathrm{2}}…
Question Number 55779 by gunawan last updated on 04/Mar/19 $$\mathrm{If}\:\int\:{x}\:\mathrm{log}\:\left(\mathrm{1}+\frac{\mathrm{1}}{{x}}\right){dx}\: \\ $$$$\:\:\:\:\:\:=\:{f}\left({x}\right)\:\centerdot\:\mathrm{log}\:\left({x}+\mathrm{1}\right)+{g}\left({x}\right)\:\centerdot\:{x}^{\mathrm{2}} +{Ax}+{C}, \\ $$$$\mathrm{then} \\ $$ Commented by gunawan last updated on 04/Mar/19 $$\mathrm{Find}\:{A}…
Question Number 55778 by gunawan last updated on 04/Mar/19 $$\int\:\:\frac{\mathrm{2}{x}−\mathrm{3}}{\left({x}^{\mathrm{2}} +{x}+\mathrm{1}\right)^{\mathrm{2}} }\:{dx}\:= \\ $$ Answered by MJS last updated on 04/Mar/19 $$\int\frac{\mathrm{2}{x}−\mathrm{3}}{\left({x}^{\mathrm{2}} +{x}+\mathrm{1}\right)^{\mathrm{2}} }{dx}= \\…
Question Number 55777 by gunawan last updated on 04/Mar/19 $$\underset{\:\mathrm{0}} {\overset{\pi} {\int}}\:\:\mathrm{cos}^{\mathrm{7}} {x}\:\mathrm{sin}\:\mathrm{3}{x}\:{dx}\:=\:\mathrm{0} \\ $$ Answered by tanmay.chaudhury50@gmail.com last updated on 04/Mar/19 $${I}=\int_{\mathrm{0}} ^{\pi} \left\{{cos}\left(\pi−{x}\right)\right\}^{\mathrm{7}}…
Question Number 55776 by gunawan last updated on 04/Mar/19 $$\underset{{n}\rightarrow\infty} {\mathrm{lim}}\:\:\underset{{r}=\mathrm{1}} {\overset{{n}} {\sum}}\:\left(\frac{{r}^{\mathrm{3}} }{{r}^{\mathrm{4}} +{n}^{\mathrm{4}} }\right)\:= \\ $$ Answered by tanmay.chaudhury50@gmail.com last updated on 04/Mar/19…
Question Number 55775 by gunawan last updated on 04/Mar/19 $$\:\underset{\:\mathrm{0}} {\overset{\mathrm{1}} {\int}}\:\:\mid\mathrm{sin}\:\mathrm{2}\pi\:{x}\mid\:{dx}\:= \\ $$ Commented by maxmathsup by imad last updated on 04/Mar/19 $${let}\:{I}\:=\int_{\mathrm{0}} ^{\mathrm{1}}…
Question Number 55774 by gunawan last updated on 04/Mar/19 $$\:\underset{−\mathrm{1}/\mathrm{2}} {\overset{\mathrm{1}/\mathrm{2}} {\int}}\:\mathrm{cos}\:{x}\:\mathrm{log}\:\left(\frac{\mathrm{1}+{x}}{\mathrm{1}−{x}}\right)\:{dx}\:= \\ $$ Answered by tanmay.chaudhury50@gmail.com last updated on 04/Mar/19 $${f}\left({x}\right)={cosxln}\left(\frac{\mathrm{1}+{x}}{\mathrm{1}−{x}}\right) \\ $$$${f}\left(−{x}\right)={cosxln}\left(\frac{\mathrm{1}−{x}}{\mathrm{1}+{x}}\right) \\…
Question Number 55773 by gunawan last updated on 04/Mar/19 $$\:\underset{−\pi} {\overset{\pi} {\int}}\:\mathrm{sin}\:{x}\:{f}\left(\mathrm{cos}\:{x}\right)\:{dx}\:= \\ $$ Answered by tanmay.chaudhury50@gmail.com last updated on 04/Mar/19 $${g}\left({x}\right)={sinxf}\left({cosx}\right) \\ $$$${g}\left(−{x}\right)=−{sinxf}\left({cosx}\right) \\…
Question Number 55772 by gunawan last updated on 04/Mar/19 $$\:\underset{−\pi/\mathrm{3}} {\overset{\pi/\mathrm{3}} {\int}}\:\frac{{x}\:\mathrm{sin}\:{x}}{\mathrm{cos}^{\mathrm{2}} {x}}\:{dx}\:= \\ $$ Commented by maxmathsup by imad last updated on 04/Mar/19 $${let}\:\:{A}=\:\int_{−\frac{\pi}{\mathrm{3}}}…