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Question Number 107924    Answers: 1   Comments: 0

Question Number 107923    Answers: 3   Comments: 0

((⊚BeMath⊚)/) ∫_0 ^1 x^(9/2) (1−x)^(5/2) dx ?

$$\:\:\:\:\:\frac{\circledcirc\mathbb{B}{e}\mathcal{M}{ath}\circledcirc}{} \\ $$$$\:\:\:\:\underset{\mathrm{0}} {\overset{\mathrm{1}} {\int}}\:{x}^{\frac{\mathrm{9}}{\mathrm{2}}} \:\left(\mathrm{1}−{x}\right)^{\frac{\mathrm{5}}{\mathrm{2}}} \:{dx}\:? \\ $$

Question Number 107922    Answers: 1   Comments: 0

if: y=(x/(x^2 +1)) then find (d(√y)/d(√x)) ?

$${if}:\:{y}=\frac{{x}}{{x}^{\mathrm{2}} +\mathrm{1}}\:{then}\:{find}\:\frac{{d}\sqrt{{y}}}{{d}\sqrt{{x}}}\:? \\ $$

Question Number 107913    Answers: 0   Comments: 0

X_0 = ((X_1 .F_(y1) + X_2 .F_(y2) + .....)/(F_(y2) + F_(y2 ) + .....)) Y_0 = ((Y_1 .F_(x1) + Y_2 .F_(x2) + ......)/(F_(x1) + F_(x2) + ......)) X_0 = ((x_1 .m_1 + x_2 .m_2 + x_3 .m_3 + ....)/(m_1 + m_2 + m_3 + ......)) Y_0 = ((y_1 .m_1 + y_2 .m_2 + y_3 .m_3 + ....)/(m_1 + m_2 + m_3 + .....)) X_z = ((l_1 .x_1 + l_2 .x_2 + l_3 .x_3 + .......)/(l_1 + l_2 + l_3 + ......)) Y_z = ((l_1 .y_1 + l_2 .y_2 + l_3 .y_3 + ...... )/(l_1 + l_2 + l_3 + ......))

$$\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\mathrm{X}_{\mathrm{0}} \:=\:\frac{\mathrm{X}_{\mathrm{1}} .\mathrm{F}_{\mathrm{y1}} \:+\:\mathrm{X}_{\mathrm{2}} .\mathrm{F}_{\mathrm{y2}} \:+\:.....}{\mathrm{F}_{\mathrm{y2}} \:+\:\mathrm{F}_{\mathrm{y2}\:} +\:.....} \\ $$$$\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\mathrm{Y}_{\mathrm{0}} \:=\:\frac{\mathrm{Y}_{\mathrm{1}} .\mathrm{F}_{\mathrm{x1}} \:+\:\mathrm{Y}_{\mathrm{2}} .\mathrm{F}_{\mathrm{x2}} \:+\:......}{\mathrm{F}_{\mathrm{x1}} \:+\:\mathrm{F}_{\mathrm{x2}} +\:......} \\ $$$$ \\ $$$$\:\:\:\:\:\:\:\mathrm{X}_{\mathrm{0}} \:=\:\frac{\mathrm{x}_{\mathrm{1}} .\mathrm{m}_{\mathrm{1}} \:+\:\mathrm{x}_{\mathrm{2}} .\mathrm{m}_{\mathrm{2}} \:+\:\mathrm{x}_{\mathrm{3}} .\mathrm{m}_{\mathrm{3}} \:+\:....}{\mathrm{m}_{\mathrm{1}} \:+\:\mathrm{m}_{\mathrm{2}} \:+\:\mathrm{m}_{\mathrm{3}} \:+\:......} \\ $$$$\:\:\:\:\:\:\:\mathrm{Y}_{\mathrm{0}} \:=\:\frac{\mathrm{y}_{\mathrm{1}} .\mathrm{m}_{\mathrm{1}} \:+\:\mathrm{y}_{\mathrm{2}} .\mathrm{m}_{\mathrm{2}} \:+\:\mathrm{y}_{\mathrm{3}} .\mathrm{m}_{\mathrm{3}} \:+\:....}{\mathrm{m}_{\mathrm{1}} \:+\:\mathrm{m}_{\mathrm{2}} \:+\:\mathrm{m}_{\mathrm{3}} \:+\:.....} \\ $$$$ \\ $$$$\:\:\:\:\:\:\:\:\mathrm{X}_{\mathrm{z}} \:=\:\frac{{l}_{\mathrm{1}} .\mathrm{x}_{\mathrm{1}} \:+\:{l}_{\mathrm{2}} \:.\mathrm{x}_{\mathrm{2}} \:+\:{l}_{\mathrm{3}} .\mathrm{x}_{\mathrm{3}} \:+\:.......}{{l}_{\mathrm{1}} \:+\:{l}_{\mathrm{2}} \:+\:{l}_{\mathrm{3}} \:+\:......} \\ $$$$\:\:\:\:\:\:\:\:\mathrm{Y}_{\mathrm{z}} \:=\:\frac{{l}_{\mathrm{1}} .{y}_{\mathrm{1}} \:+\:{l}_{\mathrm{2}} .{y}_{\mathrm{2}} \:+\:{l}_{\mathrm{3}} .{y}_{\mathrm{3}} \:+\:......\:}{{l}_{\mathrm{1}} \:+\:{l}_{\mathrm{2}} \:+\:{l}_{\mathrm{3}} \:+\:......} \\ $$$$ \\ $$

Question Number 107905    Answers: 1   Comments: 3

Which of the following set of horizontal forces would lead an object in equilibrium? (a) 5N 10N 20N (b) 6N 12N 18N (c) 8N 8N 8N (b) 2N 4N 8N 16N

$$\mathrm{W}{hich}\:{of}\:{the}\:{following}\:{set}\:{of}\:{horizontal} \\ $$$${forces}\:{would}\:{lead}\:{an}\:{object}\:{in}\:{equilibrium}? \\ $$$$\left({a}\right)\:\mathrm{5}{N}\:\:\:\mathrm{10}{N}\:\:\mathrm{20}{N} \\ $$$$\left({b}\right)\:\mathrm{6}{N}\:\:\:\mathrm{12}{N}\:\:\:\:\mathrm{18}{N} \\ $$$$\left({c}\right)\:\mathrm{8}{N}\:\:\:\mathrm{8}{N}\:\:\:\mathrm{8}{N} \\ $$$$\left({b}\right)\:\mathrm{2}{N}\:\:\:\:\mathrm{4}{N}\:\:\:\mathrm{8}{N}\:\:\:\mathrm{16}{N} \\ $$

Question Number 107900    Answers: 2   Comments: 0

((Σ BeMath Σ)/□) Given tan x−sec x = ϑ then sin x = ?

$$\:\:\:\:\:\:\:\frac{\Sigma\:\mathcal{B}{e}\mathcal{M}{ath}\:\Sigma}{\Box} \\ $$$$\:\:{Given}\:\mathrm{tan}\:{x}−\mathrm{sec}\:{x}\:=\:\vartheta\: \\ $$$$\:{then}\:\mathrm{sin}\:{x}\:=\:? \\ $$

Question Number 107883    Answers: 1   Comments: 0

Expand e^(1/x) (√(x(x+2)))

$$\mathrm{Expand}\:\mathrm{e}^{\mathrm{1}/\mathrm{x}} \sqrt{\mathrm{x}\left(\mathrm{x}+\mathrm{2}\right)} \\ $$

Question Number 107882    Answers: 1   Comments: 5

lim_(x→∞) (((a^(1/x) +b^(1/x) )/2))^x

$$\underset{\mathrm{x}\rightarrow\infty} {\mathrm{lim}}\left(\frac{\mathrm{a}^{\mathrm{1}/\mathrm{x}} +\mathrm{b}^{\mathrm{1}/\mathrm{x}} }{\mathrm{2}}\right)^{\mathrm{x}} \\ $$

Question Number 107877    Answers: 0   Comments: 0

Question Number 107867    Answers: 0   Comments: 0

Question Number 107859    Answers: 0   Comments: 0

find A_n =∫_0 ^1 x^n (√(1+x+x^2 ))dx (n natural)

$$\mathrm{find}\:\mathrm{A}_{\mathrm{n}} =\int_{\mathrm{0}} ^{\mathrm{1}} \:\mathrm{x}^{\mathrm{n}} \sqrt{\mathrm{1}+\mathrm{x}+\mathrm{x}^{\mathrm{2}} }\mathrm{dx}\:\:\left(\mathrm{n}\:\mathrm{natural}\right) \\ $$

Question Number 107858    Answers: 0   Comments: 1

calculate ∫_0 ^1 ((ln(1+x^2 ))/(1+x^4 ))dx

$$\mathrm{calculate}\:\int_{\mathrm{0}} ^{\mathrm{1}} \:\frac{\mathrm{ln}\left(\mathrm{1}+\mathrm{x}^{\mathrm{2}} \right)}{\mathrm{1}+\mathrm{x}^{\mathrm{4}} }\mathrm{dx} \\ $$

Question Number 107855    Answers: 0   Comments: 1

calculate ∫_0 ^∞ ((lnx)/((1+x)^4 ))dx

$$\mathrm{calculate}\:\int_{\mathrm{0}} ^{\infty} \:\:\frac{\mathrm{lnx}}{\left(\mathrm{1}+\mathrm{x}\right)^{\mathrm{4}} }\mathrm{dx}\: \\ $$

Question Number 107852    Answers: 2   Comments: 0

Question Number 107847    Answers: 0   Comments: 0

Question Number 107844    Answers: 0   Comments: 0

A general case: we have totally n letters, among them n_1 times A, n_2 times B, n_3 times C, n_4 times D etc. (n_1 ,n_2 ,n_3 ,n_(4,) ...≥2, n>n_1 +n_2 +n_3 +n_4 +....) how many different words can be formed using these n letters such that same letters are not next to each other. see also Q107451.

$${A}\:{general}\:{case}: \\ $$$${we}\:{have}\:{totally}\:{n}\:{letters},\:{among}\:{them} \\ $$$${n}_{\mathrm{1}} \:{times}\:{A},\:{n}_{\mathrm{2}} \:{times}\:{B},\:{n}_{\mathrm{3}} \:{times}\:{C}, \\ $$$${n}_{\mathrm{4}} \:{times}\:{D}\:{etc}. \\ $$$$\left({n}_{\mathrm{1}} ,{n}_{\mathrm{2}} ,{n}_{\mathrm{3}} ,{n}_{\mathrm{4},} ...\geqslant\mathrm{2},\:{n}>{n}_{\mathrm{1}} +{n}_{\mathrm{2}} +{n}_{\mathrm{3}} +{n}_{\mathrm{4}} +....\right) \\ $$$${how}\:{many}\:{different}\:{words}\:{can}\:{be} \\ $$$${formed}\:{using}\:{these}\:{n}\:{letters}\:{such}\:{that} \\ $$$${same}\:{letters}\:{are}\:{not}\:{next}\:{to}\:{each} \\ $$$${other}. \\ $$$$ \\ $$$${see}\:{also}\:{Q}\mathrm{107451}. \\ $$

Question Number 107839    Answers: 1   Comments: 2

App update for teachers 2.136 A new option is added to generate long division method for square root under matrix menu. Enter a number and app will generate table like below.

$$\mathrm{App}\:\mathrm{update}\:\mathrm{for}\:\mathrm{teachers}\:\mathrm{2}.\mathrm{136} \\ $$$$\mathrm{A}\:\mathrm{new}\:\mathrm{option}\:\mathrm{is}\:\mathrm{added}\:\mathrm{to}\:\mathrm{generate} \\ $$$$\mathrm{long}\:\mathrm{division}\:\mathrm{method}\:\mathrm{for}\:\mathrm{square} \\ $$$$\mathrm{root}\:\mathrm{under}\:\mathrm{matrix}\:\mathrm{menu}. \\ $$$$\mathrm{Enter}\:\mathrm{a}\:\mathrm{number}\:\mathrm{and}\:\mathrm{app}\:\mathrm{will} \\ $$$$\mathrm{generate}\:\mathrm{table}\:\mathrm{like}\:\mathrm{below}. \\ $$

Question Number 107835    Answers: 2   Comments: 0

compute In=∫_0 ^1 x^n (√(1−x)) dx

$${compute}\:{In}=\int_{\mathrm{0}} ^{\mathrm{1}} {x}^{{n}} \sqrt{\mathrm{1}−{x}}\:{dx} \\ $$

Question Number 107831    Answers: 2   Comments: 0

Question Number 107828    Answers: 3   Comments: 0

lim {((x^2 +5x+3)/(x^2 +x+2))}^x x→0 lim ((10^x −2^x −5^x +1)/(xtanx)) x→0

$${lim}\:\:\:\:\:\:\left\{\frac{{x}^{\mathrm{2}} +\mathrm{5}{x}+\mathrm{3}}{{x}^{\mathrm{2}} +{x}+\mathrm{2}}\right\}^{{x}} \\ $$$${x}\rightarrow\mathrm{0} \\ $$$${lim}\:\:\:\:\frac{\mathrm{10}^{{x}} −\mathrm{2}^{{x}} −\mathrm{5}^{{x}} +\mathrm{1}}{{xtanx}} \\ $$$${x}\rightarrow\mathrm{0} \\ $$$$ \\ $$

Question Number 107818    Answers: 2   Comments: 3

Question Number 107812    Answers: 2   Comments: 0

If the roots of the equation x^2 − x − 1 = 0 are α and β, provided that x_n = α^n + β^n . Find x_(16) .

$$\mathrm{If}\:\mathrm{the}\:\mathrm{roots}\:\mathrm{of}\:\mathrm{the}\:\mathrm{equation}\:\:\:\mathrm{x}^{\mathrm{2}} \:\:−\:\:\mathrm{x}\:\:−\:\:\mathrm{1}\:\:\:=\:\:\mathrm{0}\:\:\:\mathrm{are}\:\:\:\alpha\:\:\mathrm{and}\:\:\beta, \\ $$$$\mathrm{provided}\:\mathrm{that}\:\:\:\:\:\:\mathrm{x}_{\mathrm{n}} \:\:=\:\:\alpha^{\mathrm{n}} \:\:+\:\:\beta^{\mathrm{n}} \:\:.\:\:\:\mathrm{Find}\:\:\:\:\mathrm{x}_{\mathrm{16}} . \\ $$

Question Number 107810    Answers: 1   Comments: 0

Question Number 107794    Answers: 2   Comments: 0

Question Number 107790    Answers: 4   Comments: 5

∫_0 ^1 ln(1+x^2 )dx

$$\int_{\mathrm{0}} ^{\mathrm{1}} \mathrm{ln}\left(\mathrm{1}+\mathrm{x}^{\mathrm{2}} \right)\mathrm{dx} \\ $$

Question Number 107783    Answers: 0   Comments: 1

Σ_(n=1) ^∞ (1/(n2^n ))

$$\underset{{n}=\mathrm{1}} {\overset{\infty} {\sum}}\frac{\mathrm{1}}{{n}\mathrm{2}^{{n}} } \\ $$

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