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AllQuestion and Answers: Page 1094

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 ))

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

Question Number 107779    Answers: 1   Comments: 3

Question Number 107766    Answers: 4   Comments: 0

Question Number 107764    Answers: 0   Comments: 1

tanθ = ((Σ F_y )/(Σ F_x ))

$$\:\:\:\:\:\:\:\:\:\:\:\:\:\mathrm{tan}\theta\:=\:\:\frac{\Sigma\:\mathrm{F}_{\mathrm{y}} }{\Sigma\:\mathrm{F}_{\mathrm{x}} } \\ $$$$ \\ $$

Question Number 107756    Answers: 2   Comments: 0

((BeMath)/∐) ∫ x^2 ln (x^2 +3) dx

$$\:\frac{\mathcal{B}{e}\mathcal{M}{ath}}{\coprod} \\ $$$$\:\int\:{x}^{\mathrm{2}} \:\mathrm{ln}\:\left({x}^{\mathrm{2}} +\mathrm{3}\right)\:{dx}\: \\ $$

Question Number 107747    Answers: 3   Comments: 0

L=lim_(x→0) (((1−cos xcos 2xcos 3x)/(sin^2 2x))) = ?

$${L}=\underset{{x}\rightarrow\mathrm{0}} {\mathrm{lim}}\left(\frac{\mathrm{1}−\mathrm{cos}\:{x}\mathrm{cos}\:\mathrm{2}{x}\mathrm{cos}\:\mathrm{3}{x}}{\mathrm{sin}\:^{\mathrm{2}} \mathrm{2}{x}}\right)\:=\:? \\ $$

Question Number 107746    Answers: 1   Comments: 0

Question Number 107745    Answers: 2   Comments: 3

(i) L=lim_(x→0) [(((1+x)^(1/x) )/e)]^(1/x) = ? (ii) L=lim_(x→∞) [(x/e)−x((x/(x+1)))^x ] = ?

$$\left({i}\right)\:\:\:{L}=\underset{{x}\rightarrow\mathrm{0}} {\mathrm{lim}}\left[\frac{\left(\mathrm{1}+{x}\right)^{\mathrm{1}/{x}} }{{e}}\right]^{\mathrm{1}/{x}} \:\:=\:? \\ $$$$\left({ii}\right)\:\:{L}=\underset{{x}\rightarrow\infty} {\mathrm{lim}}\left[\frac{{x}}{{e}}−{x}\left(\frac{{x}}{{x}+\mathrm{1}}\right)^{{x}} \right]\:=\:? \\ $$

Question Number 107741    Answers: 0   Comments: 2

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