Question Number 1164 by 112358 last updated on 08/Jul/15 $${How}\:{many}\:{five}\:{digit}\:{numbers} \\ $$$${exist}\:{such}\:{that}\:{the}\:{sum}\:{of}\:{their} \\ $$$${digits}\:{equals}\:\mathrm{43}?\: \\ $$$${How}\:{many}\:{exist}\:{if}\:{the}\:{sum}\:{is} \\ $$$$\mathrm{39}? \\ $$$$ \\ $$ Commented by 123456…
Question Number 132235 by bemath last updated on 12/Feb/21 $$\:\mathrm{I}=\:\int\:\frac{\mathrm{3x}+\mathrm{5}}{\left(\mathrm{x}^{\mathrm{2}} +\mathrm{2x}+\mathrm{3}\right)^{\mathrm{2}} }\:\mathrm{dx}\:? \\ $$ Answered by bemath last updated on 12/Feb/21 Commented by bemath last…
Question Number 66696 by mathmax by abdo last updated on 18/Aug/19 $${calculate}\:{lim}_{{x}\rightarrow\mathrm{0}} \:\:\:\:\:\frac{{arctan}\left(\mathrm{1}+{x}^{\mathrm{3}} \right)−\frac{\pi}{\mathrm{4}}}{{xsin}\left({x}^{\mathrm{2}} \right)} \\ $$ Commented by kaivan.ahmadi last updated on 18/Aug/19 $${lim}_{{x}\rightarrow\mathrm{0}}…
Question Number 66697 by naka3546 last updated on 18/Aug/19 Commented by kaivan.ahmadi last updated on 18/Aug/19 $${lim}_{{x}\rightarrow\frac{\pi}{\mathrm{4}}} \:\frac{−{sin}\left({x}−\frac{\pi}{\mathrm{4}}\right)−\left(\mathrm{1}+{tan}^{\mathrm{2}} {x}\right)}{{cos}\left({x}−\frac{\pi}{\mathrm{4}}\right)}=\frac{\mathrm{0}−\left(\mathrm{1}+\mathrm{1}\right)}{\mathrm{1}}=−\mathrm{2} \\ $$ Answered by Cmr 237…
Question Number 66694 by mathmax by abdo last updated on 18/Aug/19 $$\left.{let}\:{f}\left({a}\right)\:=\int_{−\infty} ^{+\infty} \:\:\frac{{dx}}{\left({x}^{\mathrm{4}} +{x}^{\mathrm{2}} \:+{a}\right)}\:{with}\:{a}\in\right]\frac{\mathrm{1}}{\mathrm{4}},+\infty\left[\right. \\ $$$$\left.\mathrm{1}\right)\:{calculate}\:{f}\left({a}\right) \\ $$$$\left.\mathrm{2}\right){find}\:{also}\:{g}\left({a}\right)\:=\int_{−\infty} ^{+\infty} \:\frac{{dx}}{\left({x}^{\mathrm{4}} \:+{x}^{\mathrm{2}} +{a}\right)^{\mathrm{2}} }…
Question Number 66695 by mathmax by abdo last updated on 18/Aug/19 $${calculate}\:\int_{\mathrm{0}} ^{\infty} \:\:\frac{{cos}\left({arctanx}\right)}{\mathrm{4}+{x}^{\mathrm{2}} }{dx} \\ $$ Commented by ~ À ® @ 237 ~…
Question Number 1157 by navajyoti.tamuli.tamuli@gmail. last updated on 06/Jul/15 $${show}\:{that} \\ $$$${tan}^{−\mathrm{1}} \left(\frac{\sqrt{\mathrm{1}+{x}^{\mathrm{2}} }−\mathrm{1}}{{x}}\right)=\frac{\mathrm{1}}{\mathrm{2}}{tan}^{−\mathrm{1}} {x} \\ $$$$ \\ $$ Answered by prakash jain last updated…
Question Number 132230 by pticantor last updated on 12/Feb/21 $$ \\ $$$$ \\ $$$$\boldsymbol{{S}}=\underset{\boldsymbol{{k}}=\mathrm{1}} {\overset{+\infty} {\sum}}\left(\frac{\mathrm{1}}{\:\sqrt{\boldsymbol{{k}}^{\mathrm{2}} −\mathrm{1}}}−\frac{\mathrm{1}}{\:\sqrt{\boldsymbol{{k}}^{\mathrm{2}} +\mathrm{1}}}\right) \\ $$$$\boldsymbol{{S}}\:={l}\:{or}\:\boldsymbol{{S}}=\infty\:\:??? \\ $$ Answered by JDamian…
Question Number 66693 by mathmax by abdo last updated on 18/Aug/19 $${calculate}\:\:\int_{−\infty} ^{+\infty} \:\frac{{dx}}{\left({x}^{\mathrm{2}} −{x}+\mathrm{1}\right)^{\mathrm{3}} } \\ $$ Commented by mathmax by abdo last updated…
Question Number 1156 by 112358 last updated on 06/Jul/15 $${Determine}\:{the}\:{general}\:{solution} \\ $$$${of}\:{the}\:{following}\:\:{linear}\:{diophantine} \\ $$$${equation}\:{for}\:\forall{N}\in\mathbb{Z}^{+} ,{m}\in\mathbb{Z}^{+} : \\ $$$$ \\ $$$$\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\mathrm{8}{N}=\mathrm{81}{m}+\mathrm{65}\:. \\ $$ Commented by prakash…