Question Number 211578 by MrGaster last updated on 13/Sep/24 $$ \\ $$$$\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\int_{\mathrm{0}} ^{+\infty} \frac{\boldsymbol{{x}}}{\:\sqrt{\mathrm{1}+\boldsymbol{{x}}^{\mathrm{4}} }}\boldsymbol{{dx}}. \\ $$$$ \\ $$ Answered by lepuissantcedricjunior last updated on…
Question Number 211579 by MrGaster last updated on 13/Sep/24 $$ \\ $$$$\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\int\frac{\mathrm{1}}{\left(\mathrm{1}−\boldsymbol{{x}}^{\mathrm{4}} \right)\sqrt{\mathrm{1}+\boldsymbol{{x}}^{\mathrm{2}} }}\boldsymbol{{dx}}. \\ $$$$ \\ $$ Answered by lepuissantcedricjunior last updated on 13/Sep/24…
Question Number 211574 by MrGaster last updated on 13/Sep/24 $$ \\ $$$$\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\begin{cases}{\boldsymbol{{x}}^{\mathrm{2}} +\boldsymbol{\mathrm{y}}^{\mathrm{2}} =\mathrm{25}}\\{\boldsymbol{{x}}+\mathrm{2}\boldsymbol{\mathrm{y}}−\mathrm{3}=\mathrm{0}}\end{cases} \\ $$$$\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\: \\ $$$$ \\ $$$$ \\ $$ Answered by Rasheed.Sindhi…
Question Number 211575 by BaliramKumar last updated on 13/Sep/24 Answered by som(math1967) last updated on 13/Sep/24 $$\:\left(\boldsymbol{{a}}\right)\frac{\mathrm{1}}{\mathrm{2}}\left(\boldsymbol{{m}}+\boldsymbol{{n}}+\frac{\mathrm{1}}{\boldsymbol{{m}}}+\frac{\mathrm{1}}{\boldsymbol{{n}}}\right) \\ $$ Answered by A5T last updated on…
Question Number 211570 by liuxinnan last updated on 13/Sep/24 $$\left\{{x}_{{n}} \right\}>\mathrm{0} \\ $$$$\underset{{x}\rightarrow\mathrm{0}} {\mathrm{lim}}{x}_{{n}} ^{\frac{\mathrm{1}}{{n}}} ={a} \\ $$$${prove}\:{when}\:{a}<\mathrm{1}\: \\ $$$$\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\underset{{x}\rightarrow\mathrm{0}} {\mathrm{lim}}{x}_{{n}} =\mathrm{0} \\ $$$${when}\:{a}>\mathrm{1}\:\underset{{x}\rightarrow\mathrm{0}} {\mathrm{lim}}{x}_{{n}}…
Question Number 211564 by MrGaster last updated on 13/Sep/24 $$ \\ $$$$\:\:\:\:\:\:\:\:\:\:\:\mathrm{certificate}:\underset{\boldsymbol{{x}}=\mathrm{1}} {\overset{\mathrm{90}} {\sum}}\frac{\boldsymbol{\mathrm{sin}}\left(\boldsymbol{{x}}^{\mathrm{2}} \right)}{\boldsymbol{\mathrm{sin}}\left(\boldsymbol{{x}}\right)}=\mathrm{45} \\ $$$$ \\ $$$$ \\ $$ Commented by MrGaster last…
Question Number 211566 by MrGaster last updated on 13/Sep/24 $$\:\:\:\: \\ $$$$\:\:\:\:\:\:\:\:\:\:\:\mathrm{certificate}: \\ $$$$\:\:\:\:\:\:\:\:\underset{\boldsymbol{{n}}=−\infty} {\overset{+\infty} {\sum}}\frac{\mathrm{1}}{\boldsymbol{{n}}^{\mathrm{4}} +\boldsymbol{{a}}^{\mathrm{4}} }=\frac{\boldsymbol{\pi}}{\:\sqrt{\mathrm{2}}\boldsymbol{{a}}^{\mathrm{3}} }\:\frac{\boldsymbol{\mathrm{si}{n}\mathrm{h}}\left(\sqrt{\mathrm{2}}\boldsymbol{{a}\pi}\right)+\boldsymbol{\mathrm{sin}}\left(\sqrt{\mathrm{2}}\boldsymbol{{a}\pi}\right)}{\:\sqrt{\mathrm{2}}\boldsymbol{{a}}^{\mathrm{3}} \boldsymbol{\mathrm{cos}{h}}\left(\sqrt{\mathrm{2}}\boldsymbol{{a}\pi}\right)−\boldsymbol{\mathrm{cos}}\left(\sqrt{\mathrm{2}}\boldsymbol{{a}\pi}\right)} \\ $$$$ \\ $$ Terms…
Question Number 211567 by liuxinnan last updated on 13/Sep/24 $${if}\:\underset{{x}\rightarrow\mathrm{0}} {\mathrm{lim}}{f}\left({x}\right)=\underset{{x}\rightarrow\mathrm{0}} {\mathrm{lim}}{g}\left({x}\right)=\mathrm{0} \\ $$$${when}\:{do}\:{not}\:{use}\:{f}\left({x}\right)\:{to}\:\:{replace}\:{g}\left({x}\right)\:\:\: \\ $$ Commented by MrGaster last updated on 13/Sep/24 $$\mathrm{1}.\mathrm{when}\underset{{x}\rightarrow\mathrm{0}} {\mathrm{lim}}{f}\left({x}\right)=\underset{{x}\rightarrow\mathrm{0}}…
Question Number 211560 by MrGaster last updated on 13/Sep/24 $$\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\: \\ $$$$\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\mathrm{certificate}: \\ $$$$\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\boldsymbol{{I}}=\int_{\mathrm{0}} ^{\frac{\boldsymbol{\pi}}{\mathrm{2}}} \sqrt{\sqrt{\boldsymbol{{x}}^{\mathrm{2}} +\boldsymbol{\mathrm{ln}}^{\mathrm{2}} \boldsymbol{\mathrm{cos}}\left(\boldsymbol{{x}}\right)−\boldsymbol{\mathrm{lncos}}\left(\boldsymbol{{x}}\right)\boldsymbol{{dx}}}}=\frac{\boldsymbol{\pi}}{\mathrm{2}}\sqrt{\mathrm{2}\boldsymbol{\mathrm{ln}}\mathrm{2}} \\ $$$$ \\ $$ Terms of Service…
Question Number 211562 by MrGaster last updated on 13/Sep/24 $$\: \\ $$$$\:\:\:\:\:\:\:\:\:\:\:\begin{cases}{\boldsymbol{{x}}^{\mathrm{2}} +\boldsymbol{\mathrm{y}}^{\mathrm{2}} =\mathrm{1}}\\{\boldsymbol{{x}}^{\mathrm{3}} −\boldsymbol{\mathrm{y}}=\mathrm{0}}\end{cases} \\ $$ Answered by Rasheed.Sindhi last updated on 13/Sep/24 $$\:\:\:\:\:\:\:\:\:\:\:\begin{cases}{\boldsymbol{{x}}^{\mathrm{2}}…