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Question Number 63076 Answers: 0 Comments: 0
$${show}\:{that}\:{f}:{A}\rightarrow{B}\:{is}\:{bijection}\:{then}\:{f}\left({A}_{\mathrm{1}} ^{{c}} \right)=\left[{f}\left({A}_{\mathrm{1}} \right)\right]^{{c}} \\ $$
Question Number 63059 Answers: 0 Comments: 0
Question Number 63054 Answers: 0 Comments: 0
$${if}\:\Sigma\mid{a}_{{n}} \:\mid\:{is}\:{convergent},\:{then} \\ $$$${prove}\:{that}\:{there}\:{exists}\: \\ $$$${a}\:{subsequence}\:\left\{{n}_{{k}} {a}_{{n}_{{k}} } \right\}\:\:{with} \\ $$$$\underset{{k}\rightarrow\infty} {\mathrm{lim}}{n}_{{k}} {a}_{{n}_{{k}} } =\mathrm{0} \\ $$
Question Number 63034 Answers: 0 Comments: 0
$${calculate}\:\int_{\mathrm{0}} ^{\frac{\pi}{\mathrm{2}}} \left({ln}\left({cosx}\right)\right)^{\mathrm{2}} \:{dx}\: \\ $$
Question Number 63033 Answers: 0 Comments: 1
$${calculate}\:\int_{\mathrm{0}} ^{\infty} \:\:\frac{{sin}^{\mathrm{2}} \left({x}\right)}{{x}^{\mathrm{2}} \left(\mathrm{1}+{x}^{\mathrm{2}} \right)}{dx}\: \\ $$
Question Number 63032 Answers: 0 Comments: 1
$${let}\:{f}\left({z}\right)\:=\frac{\mathrm{1}}{{sin}\left(\pi{z}\right)}\:\:{calculate}\:{Res}\left({f},{n}\right)\:{with}\:{n}\:{integr} \\ $$
Question Number 63031 Answers: 0 Comments: 2
$${let}\:{f}\left({z}\right)\:=\frac{{sin}\left({z}\right)}{{z}^{\mathrm{2}} }\:\:{calculate}\:{Res}\left({f},\mathrm{0}\right) \\ $$
Question Number 63026 Answers: 0 Comments: 0
$${calculate}\:\int_{\mathrm{0}} ^{\pi} \:\:\frac{{sin}\left(\mathrm{2}{x}\right)}{\mathrm{2}{cosx}\:−\mathrm{3}{sinx}}{dx} \\ $$
Question Number 63023 Answers: 0 Comments: 1
$${calculate}\:\int_{−\infty} ^{+\infty} \:\:\:\frac{{x}^{\mathrm{2}} −\mathrm{3}}{{x}^{\mathrm{4}} \:+{x}^{\mathrm{2}} \:+\mathrm{1}}{dx}\:. \\ $$
Question Number 63021 Answers: 2 Comments: 3
$${solve}\:{this}\:{equation} \\ $$$$ \\ $$$$\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:{x}^{{y}} ={y}^{{x}} \\ $$$$ \\ $$$$ \\ $$$${x},{y}\in\mathbb{R}. \\ $$
Question Number 63017 Answers: 0 Comments: 0
$$\mathrm{445}\boldsymbol{{x}}\frac{\mathrm{5}\boldsymbol{{x}}}{} \\ $$$$ \\ $$
Question Number 63016 Answers: 1 Comments: 0
$$\mathrm{The}\:\mathrm{sides}\:\mathrm{of}\:\mathrm{a}\:\mathrm{hexagon}\:\mathrm{are}\:\mathrm{enlarged}\:\mathrm{by}\: \\ $$$$\mathrm{three}\:\mathrm{times}.\:\mathrm{Find}\:\mathrm{the}\:\mathrm{ratio}\:\mathrm{of}\:\mathrm{the}\:\mathrm{areas} \\ $$$$\mathrm{of}\:\mathrm{the}\:\mathrm{new}\:\mathrm{and}\:\mathrm{old}\:\mathrm{hexagon} \\ $$
Question Number 63015 Answers: 0 Comments: 0
$${f}\left({x},{y},{z}\right)=\:{x}\left({p}+{z}\right)+{y}\left({p}−{z}\right) \\ $$$$\:\:\:\:+\frac{\mathrm{4}{x}^{\mathrm{3}} }{{p}+{z}}+\frac{\mathrm{4}{y}^{\mathrm{3}} }{{p}−{z}}+\mathrm{4}\left({x}+{y}\right)^{\mathrm{2}} \left({y}−{x}\right) \\ $$$$\forall\:\:{p}\left({x},{y}\right)={c}+\left({x}−{y}\right)\sqrt{\mathrm{1}+\left({x}+{y}\right)^{\mathrm{2}} } \\ $$$$\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:+\left({x}^{\mathrm{2}} −{y}^{\mathrm{2}} \right) \\ $$$${Determine}\:{x},{y},{z}\:{such}\:{that}\:{f}\:{is} \\ $$$${maximum}.\:\left({c}\:{is}\:{a}\:{constant}\right). \\ $$$${Assume}\:{y}\geqslant{x}. \\ $$
Question Number 62998 Answers: 0 Comments: 12
$${Solve}\:{for}\:{x}:\:\:\mathrm{5}^{\boldsymbol{{x}}} +\mathrm{6}\boldsymbol{{x}}=\mathrm{7} \\ $$
Question Number 62997 Answers: 0 Comments: 1
$$\int\frac{{ln}\left(\mathrm{1}+{xsin}^{\mathrm{2}} \left({x}\right)\right)}{{sin}^{\mathrm{2}} \left({x}\right)}\:{dx} \\ $$
Question Number 62995 Answers: 0 Comments: 0
Question Number 62987 Answers: 0 Comments: 0
Question Number 62983 Answers: 3 Comments: 0
$$\mathrm{If}\:\:\:\mathrm{tan}\:\mathrm{2}\theta\:\mathrm{tan}\:\theta\:=\:\mathrm{1},\:\mathrm{then}\:\theta\:= \\ $$
Question Number 62982 Answers: 1 Comments: 0
$$\frac{\boldsymbol{\mathrm{tg}}\left(\frac{\boldsymbol{\mathrm{x}}}{\mathrm{2}}\right)−\mathrm{1}}{\boldsymbol{\mathrm{ctg}}\left(\frac{\boldsymbol{\mathrm{x}}}{\mathrm{2}}\right)}=\mathrm{2}\boldsymbol{\mathrm{sin}}\left(\frac{\boldsymbol{\mathrm{x}}}{\mathrm{2}}\right)\:\:\boldsymbol{\mathrm{x}}\in\left(\mathrm{180}°;\mathrm{540}°\right) \\ $$$$ \\ $$
Question Number 62981 Answers: 1 Comments: 1
Question Number 62970 Answers: 1 Comments: 1
Question Number 62945 Answers: 1 Comments: 0
$$\mathrm{Find}\:\mathrm{the}\:\mathrm{greatest}\:\mathrm{coefficient}\:\mathrm{in}\:\mathrm{the}\:\mathrm{expansion}\:\mathrm{of}\:\:\:\left(\mathrm{6}\:−\:\mathrm{4x}\right)^{−\mathrm{3}} \\ $$
Question Number 62942 Answers: 1 Comments: 10
$$\mathrm{Make}\:\:\mathrm{r}\:\:\mathrm{the}\:\mathrm{subject}\:\mathrm{of}\:\mathrm{the}\:\mathrm{formular}: \\ $$$$\:\:\:\:\:\:\:\:\:\:\:\mathrm{S}\:\:=\:\:\frac{\mathrm{a}\left(\mathrm{r}^{\mathrm{n}} \:−\:\mathrm{1}\right)}{\mathrm{r}\:−\:\mathrm{1}} \\ $$
Question Number 62938 Answers: 1 Comments: 1
Question Number 62937 Answers: 1 Comments: 3
$$\int_{\mathrm{0}} ^{\:\:\mathrm{x}} \frac{\mathrm{1}}{\mathrm{1}+\mathrm{x}^{\mathrm{2}} }\:\mathrm{dx} \\ $$
Question Number 62930 Answers: 0 Comments: 0
$${find}\:{the}\:{value}\:\int_{\mathrm{0}} ^{\mathrm{1}} \:{x}^{\sqrt{{x}}} {dx}\:\left({study}\:{first}\:{the}\:{convergence}\right) \\ $$
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