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Question Number 221025 by Tawa11 last updated on 22/May/25 Answered by mr W last updated on 23/May/25 $$\mathrm{20}\:{convex}\:{quadrilaterals}\:\left({red}\right) \\ $$$$\mathrm{5}\:{concave}\:{quadrilaterals}\:\left({blue}\right) \\ $$$$\mathrm{25}\:{cross}−{quadrilaterals}\:\left({green}\right) \\ $$$$\Rightarrow{totally}\:\mathrm{50}\:{quadrilaterals} \\…
Question Number 220987 by fantastic last updated on 21/May/25 $${Let}\:{a},{b},{c}\:{be}\:{positive}\:{reals}\:{such}\:{that}\:{abc}=\mathrm{1}.{prove}\:{that} \\ $$$$\frac{\mathrm{1}}{{a}^{\mathrm{3}} \left({b}+{c}\right)}+\frac{\mathrm{1}}{{b}^{\mathrm{3}} \left({c}+{a}\right)}+\frac{\mathrm{1}}{{c}^{\mathrm{3}} \left({a}+{b}\right)}\geqslant\frac{\mathrm{3}}{\mathrm{2}} \\ $$ Answered by mr W last updated on 21/May/25…
Question Number 220976 by SdC355 last updated on 21/May/25 $$\int_{−\infty} ^{\:+\infty} \int_{−\infty} ^{\:+\infty} \:{e}^{−\left({x}^{\mathrm{2}} +{y}^{\mathrm{2}} \right)} \mathrm{d}{V}\:\rightarrow\int_{\mathrm{0}} ^{\:\mathrm{2}\pi} \int_{\mathrm{0}} ^{\:\infty} \:{r}\centerdot{e}^{−{r}^{\mathrm{2}} } \mathrm{d}{r}\mathrm{d}\theta=\pi \\ $$$$\mathrm{i}\:\mathrm{can}'\mathrm{t}\:\mathrm{understand}\:\mathrm{domain}\:\mathrm{of}\:\mathrm{integration}…
Question Number 220972 by SdC355 last updated on 21/May/25 $$\int_{−\infty} ^{+\infty} \int_{−\infty} ^{+\infty} \:\:\:\frac{\mathrm{1}}{\left({x}^{\mathrm{2}} +{y}^{\mathrm{2}} +\mathrm{4}\right)^{\mathrm{2}} }\:\mathrm{d}{x}\mathrm{d}{y} \\ $$$${x}={r}\mathrm{cos}\left(\theta\right) \\ $$$${y}={r}\mathrm{sin}\left(\theta\right) \\ $$$$\mid\mid\boldsymbol{{J}}\mid\mid={r}\mathrm{d}{r}\mathrm{d}\theta \\ $$$$\int_{\mathrm{0}}…
Question Number 220973 by MrGaster last updated on 21/May/25 $$\mathrm{Prove}:\underset{{k}=\mathrm{0}} {\overset{{n}} {\sum}}\left(−\frac{\mathrm{1}}{\mathrm{3}}\right)^{{k}} \mathrm{cos}^{\mathrm{3}} \left(\mathrm{3}^{{k}−{n}} \pi\right)=\frac{\mathrm{3}}{\mathrm{4}}\left[\left(−\frac{\mathrm{1}}{\mathrm{3}}\right)^{{n}+\mathrm{1}} +\mathrm{cos}\frac{\pi}{\mathrm{3}^{{n}} }\right] \\ $$ Answered by universe last updated on…
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Question Number 220971 by Tawa11 last updated on 21/May/25 Commented by mr W last updated on 21/May/25 $$\mathrm{40}? \\ $$ Commented by Frix last updated…
Question Number 220964 by fantastic last updated on 21/May/25 $${Find}\:{the}\:{general}\:{solution}\:{of}\:{the}\:{differential}\:{equation} \\ $$$${x}^{\mathrm{2}} \:\frac{{d}^{\mathrm{3}} {y}}{{dx}^{\mathrm{3}} }\:+\:{x}\frac{{d}^{\mathrm{2}} {y}}{{dx}^{\mathrm{2}} }−\mathrm{6}\frac{{dy}}{{dx}}+\mathrm{6}\frac{{y}}{{x}}=\frac{{x}\:\mathrm{ln}\:{x}+\mathrm{1}}{{x}^{\mathrm{2}} },\left[{x}>\mathrm{0}\right] \\ $$ Terms of Service Privacy Policy…
Question Number 220963 by fantastic last updated on 21/May/25 $${Let}\:{f}:\mathbb{R}^{\mathrm{2}} \rightarrow\mathbb{R}\:{be}\:{defined}\:{by}\:{f}\left({x},{y}\right)=\left\{\frac{{y}}{\underset{\:\:\mathrm{1},\:{y}=\mathrm{0}} {\mathrm{sin}\:{y}}},\:{y}\neq\mathrm{0}\right. \\ $$$${Then}\:{the}\:{integral}\:\frac{\mathrm{1}}{\pi^{\mathrm{2}} }\underset{{x}=\mathrm{0}} {\overset{\mathrm{1}} {\int}}\underset{{y}=\mathrm{sin}^{−\mathrm{1}} {x}} {\overset{\frac{\pi}{\mathrm{2}}} {\int}}{f}\left({x},{y}\right){dy}\:{dx}\:{correct}\:{upto}\:{three}\:{decimal}\:{places},{is}… \\ $$ Answered by MrGaster…