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Question Number 158768 Answers: 1 Comments: 0
$$ \\ $$$$\mathrm{evaluate} \\ $$$$\int\mathrm{2x}\sqrt{\mathrm{4x}−\mathrm{5}}\:\mathrm{dx} \\ $$
Question Number 158761 Answers: 0 Comments: 0
Question Number 158760 Answers: 1 Comments: 2
$${f}\left({x}\right)=\left[{sgn}\left({x}^{\mathrm{2}} −\mathrm{1}\right)+{sgn}\left(\mathrm{sin}\:\pi{x}\right)\right] \\ $$$${faind}\:\:\:\underset{{x}\rightarrow\mathrm{1}} {\mathrm{lim}}{f}\left({x}\right)=? \\ $$
Question Number 158759 Answers: 1 Comments: 0
$$\mathrm{Prove}\:\mathrm{that}: \\ $$$$\underset{\boldsymbol{\mathrm{n}}\rightarrow\infty} {\mathrm{lim}}\sqrt[{\boldsymbol{\mathrm{n}}}]{\underset{\boldsymbol{\mathrm{k}}=\mathrm{0}} {\overset{\mathrm{2}\boldsymbol{\mathrm{n}}} {\sum}}\left(-\mathrm{1}\right)^{\boldsymbol{\mathrm{k}}} \:\centerdot\:\frac{\mathrm{4n}\:+\:\mathrm{1}}{\mathrm{4n}\:-\:\mathrm{2k}\:+\:\mathrm{1}}\begin{pmatrix}{\mathrm{2n}}\\{\:\mathrm{k}}\end{pmatrix}}\:=\:\mathrm{1} \\ $$$$ \\ $$
Question Number 158751 Answers: 0 Comments: 0
$${Q}\:\mathrm{158528} \\ $$$$ \\ $$$$\:\:\:\:\:\:\mathbb{P}=\underset{{n}=\mathrm{1}} {\overset{\infty} {\prod}}\left(\frac{\left({n}+\mathrm{1}\right)^{\mathrm{3}} −\mathrm{1}}{\left({n}+\mathrm{1}\right)^{\mathrm{3}} +\mathrm{1}}\right) \\ $$$$\Rightarrow\:\mathbb{P}\:=\:\underset{{n}=\mathrm{1}} {\overset{\infty} {\prod}}\left(\frac{\left({n}+\mathrm{1}\right)^{\mathrm{3}} −\mathrm{1}^{\mathrm{3}} }{\left({n}+\mathrm{1}\right)^{\mathrm{3}} +\mathrm{1}^{\mathrm{3}} }\right) \\ $$$$\Rightarrow\:\mathbb{P}\:=\:\underset{{n}=\mathrm{1}} {\overset{\infty} {\prod}}\left\{\frac{\left({n}+\mathrm{1}−\mathrm{1}\right)\left({n}^{\mathrm{2}} +\mathrm{2}{n}+\mathrm{1}+{n}+\mathrm{1}+\mathrm{1}\right)}{\left({n}+\mathrm{1}+\mathrm{1}\right)\left({n}^{\mathrm{2}} +\mathrm{2}{n}+\mathrm{1}−{n}−\mathrm{1}+\mathrm{1}\right)}\right\} \\ $$$$\Rightarrow\:\mathbb{P}\:=\:\underset{{n}=\mathrm{1}} {\overset{\infty} {\prod}}\left\{\frac{{n}}{{n}+\mathrm{2}}\right\}\bullet\underset{{n}=\mathrm{1}} {\overset{\infty} {\prod}}\left\{\frac{{n}^{\mathrm{2}} +\mathrm{3}{n}+\mathrm{3}}{{n}^{\mathrm{2}} +{n}+\mathrm{1}}\right\} \\ $$$$=\:\underset{{n}\rightarrow\infty} {\mathrm{lim}}\underset{{k}=\mathrm{1}} {\overset{{n}} {\prod}}\left\{\frac{{k}}{{k}+\mathrm{2}}\right\}\bullet\underset{{n}\rightarrow\infty} {\mathrm{lim}}\underset{{k}=\mathrm{1}} {\overset{{n}} {\prod}}\left\{\frac{{k}^{\mathrm{2}} +\mathrm{3}{k}+\mathrm{3}}{{k}^{\mathrm{2}} +{k}+\mathrm{1}}\right\} \\ $$$$=\:\underset{{n}\rightarrow\infty} {\mathrm{lim}}\left\{\frac{\mathrm{1}}{\mathrm{3}}\bullet\frac{\mathrm{2}}{\mathrm{4}}\bullet\frac{\mathrm{3}}{\mathrm{5}}\bullet...\bullet\frac{{n}}{{n}+\mathrm{2}}\right\}×\underset{{n}\rightarrow\infty} {\mathrm{lim}}\left\{\frac{\mathrm{7}}{\mathrm{3}}\bullet\frac{\mathrm{13}}{\mathrm{7}}\bullet...\bullet\frac{{n}^{\mathrm{2}} +\mathrm{3}{n}+\mathrm{3}}{{n}^{\mathrm{2}} +{n}+\mathrm{1}}\right\} \\ $$$$=\mathrm{2}\underset{{n}\rightarrow\infty} {\mathrm{lim}}\left\{\frac{\mathrm{1}}{\left({n}+\mathrm{1}\right)\left({n}+\mathrm{2}\right)}\right\}×\frac{\mathrm{1}}{\mathrm{3}}\underset{{n}\rightarrow\infty} {\mathrm{lim}}\left\{{n}^{\mathrm{2}} +\mathrm{3}{n}+\mathrm{3}\right\} \\ $$$$=\frac{\mathrm{2}}{\mathrm{3}}\underset{{n}\rightarrow\infty} {\mathrm{lim}}\left\{\frac{{n}^{\mathrm{2}} +\mathrm{3}{n}+\mathrm{3}}{{n}^{\mathrm{2}} +\mathrm{3}{n}+\mathrm{2}}\right\}\:=\:\frac{\mathrm{2}}{\mathrm{3}}\underset{{n}\rightarrow\infty} {\mathrm{lim}}\left\{\frac{\mathrm{1}+\frac{\mathrm{3}}{{n}}+\frac{\mathrm{3}}{{n}^{\mathrm{2}} }}{\mathrm{1}+\frac{\mathrm{3}}{{n}}+\frac{\mathrm{2}}{{n}^{\mathrm{2}} }}\right\}\:=\:\frac{\mathrm{2}}{\mathrm{3}}. \\ $$$$ \\ $$$$\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\mathbb{P}\:=\:\underset{{n}=\mathrm{1}} {\overset{\infty} {\prod}}\left(\frac{\left({n}+\mathrm{1}\right)^{\mathrm{3}} −\mathrm{1}}{\left({n}+\mathrm{1}\right)^{\mathrm{3}} +\mathrm{1}}\right)\:=\:\frac{\mathrm{2}}{\mathrm{3}}.. \\ $$$$ \\ $$$$\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:...............\mathscr{L}{e}\:{puissant}............... \\ $$
Question Number 158749 Answers: 2 Comments: 1
Question Number 158742 Answers: 1 Comments: 1
Question Number 158740 Answers: 1 Comments: 1
Question Number 158735 Answers: 0 Comments: 2
Question Number 158731 Answers: 0 Comments: 0
Question Number 158708 Answers: 2 Comments: 1
Question Number 158707 Answers: 0 Comments: 0
Question Number 158704 Answers: 0 Comments: 0
Question Number 158724 Answers: 2 Comments: 0
$$\mathrm{let}\:\:\mathrm{a}>\mathrm{b}>\mathrm{c}>\mathrm{0}\:\:\mathrm{solve}\:\mathrm{in}\:\mathbb{R} \\ $$$$\begin{cases}{\mathrm{ax}\:+\:\mathrm{by}\:+\:\mathrm{cz}\:=\:\mathrm{a}}\\{\mathrm{bx}\:+\:\mathrm{cy}\:+\:\mathrm{az}\:=\:\mathrm{b}}\\{\mathrm{cx}\:+\:\mathrm{ay}\:+\:\mathrm{bz}\:=\:\mathrm{c}}\end{cases} \\ $$$$ \\ $$
Question Number 158700 Answers: 0 Comments: 0
Question Number 158699 Answers: 0 Comments: 3
$$\sqrt{\left(\mathrm{log}_{\mathrm{3}} \mathrm{3}\sqrt{\mathrm{3x}}+\mathrm{log}_{\mathrm{x}} \mathrm{3}\sqrt{\mathrm{3x}}\right)\mathrm{log}_{\mathrm{3}} \mathrm{x}^{\mathrm{3}} }+\sqrt{\left(\frac{\mathrm{log}_{\mathrm{3}} \mathrm{3}\sqrt{\mathrm{x}}}{\mathrm{3}}+\frac{\mathrm{log}_{\mathrm{x}} \mathrm{3}\sqrt{\mathrm{x}}}{\mathrm{3}}\right)\mathrm{log}_{\mathrm{3}} \mathrm{x}^{\mathrm{3}} }=\mathrm{2} \\ $$$$\mathrm{please}\:\mathrm{i}\:\mathrm{need}\:\mathrm{help}. \\ $$$$\mathrm{Ive}\:\mathrm{been}\:\mathrm{trying}\:\mathrm{but}\:\mathrm{still}\:\mathrm{not}\:\mathrm{getting}\: \\ $$$$\mathrm{answer}. \\ $$
Question Number 158698 Answers: 0 Comments: 0
$$\left({x}+\mathrm{1}\right)^{\mathrm{2}} \left({x}^{\mathrm{2}} −\mathrm{4}{c}^{\mathrm{2}} \right)=\mathrm{4}{c}^{\mathrm{4}} \\ $$
Question Number 158697 Answers: 1 Comments: 0
$$\int_{\mathrm{0}} ^{\mathrm{1}} {ln}\left({ln}\left(\mathrm{1}−{x}\right)\right){dx}=? \\ $$
Question Number 158696 Answers: 0 Comments: 0
$${find}\:{the}\:{partial}\:{differention}\:{equation}\:{if}\: \\ $$$${u}\:=\:\left({g}\:{o}\:{f}\:\right)\:\left({x}+{y}\right) \\ $$$$ \\ $$$${help}\:{me}\:{sir} \\ $$
Question Number 158691 Answers: 1 Comments: 0
$$ \\ $$$$\:\:\:\:\:\underset{{n}=\mathrm{1}} {\overset{\infty} {\sum}}\frac{\:\left(−\mathrm{1}\right)^{\:{n}} \:\mathcal{H}_{\:\mathrm{2}{n}} }{\mathrm{2}{n}}\:=? \\ $$
Question Number 158687 Answers: 1 Comments: 0
$$\forall{x}\in{R}\:\:{f}\left({x}\right)={f}\:'\:\left({x}\right)\:\:\:\:\:\:{and}\:\:\:{f}\left(\mathrm{0}\right)=\mathrm{1} \\ $$$${prove}\:\:{f}\left({a}+{b}\right)={f}\left({a}\right)×{f}\left({b}\right) \\ $$
Question Number 158675 Answers: 2 Comments: 1
Question Number 158674 Answers: 0 Comments: 2
Question Number 158671 Answers: 0 Comments: 3
$$\forall{x}\in{R}\:;\:{f}\left({x}\right)={f}\:'\left({x}\right) \\ $$$${prove}\:{f}\left({x}+{y}\right)={f}\left({x}\right){f}\left({y}\right) \\ $$
Question Number 158668 Answers: 1 Comments: 0
$$\:{f}\left({f}\left({x}\right)\right)=\:\left(\mathrm{9}{x}^{\mathrm{2}} +\mathrm{6}{x}+\mathrm{2}\right){f}\left({x}\right) \\ $$$$\:{f}\left({x}\right)=? \\ $$
Question Number 158664 Answers: 2 Comments: 3
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