Question Number 137461 by mathlove last updated on 03/Apr/21 Answered by Ñï= last updated on 03/Apr/21 $$\int\frac{\mathrm{1}+{x}^{\mathrm{2}} }{\mathrm{2}{x}+{x}^{\mathrm{3}} }{dx}=\frac{\mathrm{1}}{\mathrm{3}}\int\frac{\mathrm{2}+\mathrm{3}{x}^{\mathrm{2}} +\mathrm{1}}{\mathrm{2}{x}+{x}^{\mathrm{3}} }{dx}=\frac{\mathrm{1}}{\mathrm{3}}{ln}\mid\mathrm{2}{x}+{x}^{\mathrm{3}} \mid+\frac{\mathrm{1}}{\mathrm{3}}\int\frac{{dx}}{{x}\left(\mathrm{2}+{x}^{\mathrm{2}} \right)} \\ $$$$=\frac{\mathrm{1}}{\mathrm{3}}{ln}\mid\mathrm{2}{x}+{x}^{\mathrm{3}}…
Question Number 71926 by TawaTawa last updated on 22/Oct/19 Commented by Prithwish sen last updated on 22/Oct/19 $$\boldsymbol{\mathrm{If}}\:\:!!\:\:\boldsymbol{\mathrm{means}}\:\boldsymbol{\mathrm{mere}}\:\boldsymbol{\mathrm{factorial}}\:\boldsymbol{\mathrm{than}}\:\mathrm{3002}!!\:\boldsymbol{\mathrm{must}} \\ $$$$\boldsymbol{\mathrm{be}}\:\boldsymbol{\mathrm{divisible}}\:\boldsymbol{\mathrm{by}}\:\mathrm{2003} \\ $$$$\boldsymbol{\mathrm{i}}.\boldsymbol{\mathrm{e}} \\ $$$$\:\:\:\:\:\mathrm{3092}!!\:\equiv\:\mathrm{0}\:\left(\boldsymbol{\mathrm{mod}}\:\mathrm{2003}\right) \\…
Question Number 6390 by FilupSmith last updated on 25/Jun/16 $$\mathrm{Solve}: \\ $$$${I}\left({n}\right)=\int_{\mathrm{0}} ^{\:{n}} \left(−\mathrm{1}\right)^{\lfloor{x}\rfloor} {x}^{\mathrm{2}} {dx} \\ $$ Commented by prakash jain last updated on…
Question Number 71925 by Henri Boucatchou last updated on 22/Oct/19 $$\:\boldsymbol{{Solve}}\:\:\sqrt{\mathrm{1}\:−\:\boldsymbol{{x}}^{\mathrm{2}} }\:=\:\boldsymbol{{x}}^{\mathrm{4}/\mathrm{3}} \\ $$ Answered by MJS last updated on 23/Oct/19 $${x}={t}^{\mathrm{3}/\mathrm{2}} \\ $$$$\sqrt{\mathrm{1}−{t}^{\mathrm{3}} }={t}^{\mathrm{2}}…
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Question Number 6384 by Rasheed Soomro last updated on 25/Jun/16 Commented by Rasheed Soomro last updated on 25/Jun/16 $${Question}\:{reposted}. \\ $$ Commented by nburiburu last…
Question Number 71918 by lalitchand last updated on 22/Oct/19 $$\mathrm{If}\:\mathrm{Cos}\theta=\frac{\mathrm{x}\:\mathrm{cos}\beta\:−\:\mathrm{y}}{\mathrm{x}\:−\:\mathrm{y}\:\mathrm{cos}\beta}\:\:\mathrm{then}\:\mathrm{prove}\:\mathrm{that},\:\mathrm{tan}\frac{\theta}{\mathrm{2}}\:=\sqrt{\frac{\mathrm{x}−\mathrm{y}}{\mathrm{x}+\mathrm{y}}\:}\:\mathrm{tan}\frac{\beta}{\mathrm{2}} \\ $$ Commented by Prithwish sen last updated on 22/Oct/19 $$\mathrm{Using}\:\mathrm{componendo}\:\mathrm{dividendo} \\ $$$$\frac{\mathrm{1}−\mathrm{cos}\theta}{\mathrm{1}+\mathrm{cos}\theta}\:=\:\frac{\left(\mathrm{1}−\mathrm{cos}\beta\right)\left(\mathrm{x}+\mathrm{y}\right)}{\left(\mathrm{1}+\mathrm{cos}\beta\right)\left(\mathrm{x}−\mathrm{y}\right)} \\ $$$$\boldsymbol{\mathrm{tan}}\frac{\boldsymbol{\theta}}{\mathrm{2}}\:=\:\sqrt{\frac{\boldsymbol{\mathrm{x}}+\boldsymbol{\mathrm{y}}}{\boldsymbol{\mathrm{x}}−\boldsymbol{\mathrm{y}}}}\:\boldsymbol{\mathrm{tan}}\frac{\boldsymbol{\beta}}{\mathrm{2}}…
Question Number 137448 by liberty last updated on 03/Apr/21 $$\int_{\mathrm{0}} ^{\:\mathrm{3}/\mathrm{4}} \frac{{dx}}{\left({x}+\mathrm{1}\right)\sqrt{{x}^{\mathrm{2}} +\mathrm{1}}}\:? \\ $$ Answered by MJS_new last updated on 03/Apr/21 $$\int\frac{{dx}}{\left({x}+\mathrm{1}\right)\sqrt{{x}^{\mathrm{2}} +\mathrm{1}}}= \\…
Question Number 6374 by sanusihammed last updated on 25/Jun/16 $${Show}\:{that} \\ $$$$\overset{−} {{a}}\:×\:\left({b}\:×\:{c}\right)\:=\:\left(\overset{−} {{a}}\:.\:\overset{−} {{c}}\right)\overset{−} {{b}}\:\:−\:\:\left(\overset{−} {{a}}\:.\:\overset{−} {{b}}\right)\overset{−} {{c}} \\ $$ Commented by FilupSmith last…