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Question Number 63251 Answers: 0 Comments: 0

∫_( 0) ^( (π/2)) sin^(−1) (m cosθ) dθ

$$\int_{\:\mathrm{0}} ^{\:\frac{\pi}{\mathrm{2}}} \:\mathrm{sin}^{−\mathrm{1}} \left(\mathrm{m}\:\mathrm{cos}\theta\right)\:\mathrm{d}\theta \\ $$

Question Number 63215 Answers: 0 Comments: 1

calculate lim_(n→+∞) {n (1+(1/n))^n −en}

$${calculate}\:{lim}_{{n}\rightarrow+\infty} \left\{{n}\:\left(\mathrm{1}+\frac{\mathrm{1}}{{n}}\right)^{{n}} −{en}\right\} \\ $$

Question Number 63214 Answers: 0 Comments: 1

calculate ∫_0 ^∞ x e^(−(x^2 /a^2 )) sin(bx)dx with a>0 and b>0

$${calculate}\:\:\int_{\mathrm{0}} ^{\infty} \:{x}\:{e}^{−\frac{{x}^{\mathrm{2}} }{{a}^{\mathrm{2}} }} \:\:{sin}\left({bx}\right){dx}\:\:{with}\:\:{a}>\mathrm{0}\:{and}\:{b}>\mathrm{0} \\ $$

Question Number 63267 Answers: 0 Comments: 3

lim_(n→∞) (((n^3 + 1)/(n^3 − 1)))^(2n − n^3 )

$$\:\:\:\:\underset{\mathrm{n}\rightarrow\infty} {\mathrm{lim}}\:\:\left(\frac{\mathrm{n}^{\mathrm{3}} \:+\:\mathrm{1}}{\mathrm{n}^{\mathrm{3}} \:−\:\mathrm{1}}\right)^{\mathrm{2n}\:−\:\mathrm{n}^{\mathrm{3}} } \\ $$

Question Number 63206 Answers: 1 Comments: 1

Question Number 63203 Answers: 0 Comments: 5

Question Number 63268 Answers: 0 Comments: 0

Question Number 63194 Answers: 1 Comments: 0

Question Number 63190 Answers: 0 Comments: 3

Test its convergence: Σ_(n = 1) ^∞ (1/(n^3 sin^2 n))

$$\mathrm{Test}\:\mathrm{its}\:\mathrm{convergence}:\:\:\:\:\:\:\:\:\underset{\mathrm{n}\:=\:\mathrm{1}} {\overset{\infty} {\sum}}\:\frac{\mathrm{1}}{\mathrm{n}^{\mathrm{3}} \:\mathrm{sin}^{\mathrm{2}} \mathrm{n}} \\ $$

Question Number 63292 Answers: 0 Comments: 4

∫_0 ^1 ∫_0 ^1 (dy/(1+y(x^2 −x))) dx

$$\int_{\mathrm{0}} ^{\mathrm{1}} \int_{\mathrm{0}} ^{\mathrm{1}} \:\frac{{dy}}{\mathrm{1}+{y}\left({x}^{\mathrm{2}} −{x}\right)}\:{dx} \\ $$

Question Number 63291 Answers: 0 Comments: 3

find some of all real x such that ((4x^2 +15x+17)/(x^2 +4x+12)) = ((5x^2 +16x+18)/(2x^2 +5x+13))

$${find}\:{some}\:{of}\:{all}\:{real}\:{x}\:{such}\:{that} \\ $$$$ \\ $$$$\frac{\mathrm{4}{x}^{\mathrm{2}} +\mathrm{15}{x}+\mathrm{17}}{{x}^{\mathrm{2}} +\mathrm{4}{x}+\mathrm{12}}\:=\:\frac{\mathrm{5}{x}^{\mathrm{2}} +\mathrm{16}{x}+\mathrm{18}}{\mathrm{2}{x}^{\mathrm{2}} +\mathrm{5}{x}+\mathrm{13}} \\ $$

Question Number 63290 Answers: 0 Comments: 0

Question Number 63289 Answers: 0 Comments: 0

Question Number 63288 Answers: 1 Comments: 0

Question Number 63218 Answers: 0 Comments: 8

Q.63108 (A check) eq. of ellipse (x^2 /4)+y^2 =1 Inscribed equilateral △ABC of side s=((16(√6))/(√(365))) Do these points satisfy for A, B, C ? A((4/(√(365))), ((19)/(√(365)))) ; B[−(((20+8(√3)))/(√(365))), ((8(√3)−5)/(√(365)))] C[−(((20−8(√3)))/(√(365))) , −(((5+8(√3)))/(√(365)))] θ=45°

$${Q}.\mathrm{63108}\:\:\:\left({A}\:{check}\right) \\ $$$${eq}.\:{of}\:{ellipse} \\ $$$$\frac{{x}^{\mathrm{2}} }{\mathrm{4}}+{y}^{\mathrm{2}} =\mathrm{1} \\ $$$${Inscribed}\:{equilateral}\:\bigtriangleup{ABC} \\ $$$${of}\:{side}\:\boldsymbol{{s}}=\frac{\mathrm{16}\sqrt{\mathrm{6}}}{\sqrt{\mathrm{365}}} \\ $$$${Do}\:{these}\:{points}\:{satisfy}\:{for} \\ $$$${A},\:{B},\:{C}\:? \\ $$$${A}\left(\frac{\mathrm{4}}{\sqrt{\mathrm{365}}},\:\frac{\mathrm{19}}{\sqrt{\mathrm{365}}}\right)\:\:\:;\:\: \\ $$$${B}\left[−\frac{\left(\mathrm{20}+\mathrm{8}\sqrt{\mathrm{3}}\right)}{\sqrt{\mathrm{365}}},\:\frac{\mathrm{8}\sqrt{\mathrm{3}}−\mathrm{5}}{\sqrt{\mathrm{365}}}\right] \\ $$$${C}\left[−\frac{\left(\mathrm{20}−\mathrm{8}\sqrt{\mathrm{3}}\right)}{\sqrt{\mathrm{365}}}\:,\:−\frac{\left(\mathrm{5}+\mathrm{8}\sqrt{\mathrm{3}}\right)}{\sqrt{\mathrm{365}}}\right] \\ $$$$\:\theta=\mathrm{45}° \\ $$

Question Number 63178 Answers: 2 Comments: 1

Question Number 63176 Answers: 1 Comments: 1

Question Number 63651 Answers: 0 Comments: 0

let S_n (x)=Σ_(k=0) ^n e^(−k) sin(k^2 x) 1) determine 2 sequence U_n (x) and V_n (x) wich verify U_n ≤ S_n ≤ V_n 2) let S =lim_(n→+∞) S(x) study the convergence of S.

$${let}\:{S}_{{n}} \left({x}\right)=\sum_{{k}=\mathrm{0}} ^{{n}} \:{e}^{−{k}} {sin}\left({k}^{\mathrm{2}} {x}\right) \\ $$$$\left.\mathrm{1}\right)\:{determine}\:\mathrm{2}\:{sequence}\:\:{U}_{{n}} \left({x}\right)\:{and}\:{V}_{{n}} \left({x}\right)\:{wich}\:{verify}\:{U}_{{n}} \leqslant\:{S}_{{n}} \leqslant\:{V}_{{n}} \\ $$$$\left.\mathrm{2}\right)\:{let}\:\:{S}\:={lim}_{{n}\rightarrow+\infty} \:{S}\left({x}\right)\:\:{study}\:{the}\:{convergence}\:{of}\:{S}. \\ $$

Question Number 63175 Answers: 0 Comments: 2

solve for x x^x^x = 16 x = 2, but how to use Lambert W function

$$\mathrm{solve}\:\mathrm{for}\:\mathrm{x}\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\mathrm{x}^{\mathrm{x}^{\mathrm{x}} } \:=\:\:\mathrm{16} \\ $$$$\mathrm{x}\:=\:\mathrm{2},\:\:\:\:\:\mathrm{but}\:\mathrm{how}\:\mathrm{to}\:\mathrm{use}\:\mathrm{Lambert}\:\mathrm{W}\:\mathrm{function} \\ $$

Question Number 63165 Answers: 0 Comments: 0

let W_n =Σ_(k=0) ^n (1/(3k+1)) determine W_n interms of H_n H_n =Σ_(k=1) ^n (1/k)

$${let}\:{W}_{{n}} =\sum_{{k}=\mathrm{0}} ^{{n}} \:\:\frac{\mathrm{1}}{\mathrm{3}{k}+\mathrm{1}}\:\:\:{determine}\:{W}_{{n}} \:{interms}\:{of}\:{H}_{{n}} \\ $$$${H}_{{n}} =\sum_{{k}=\mathrm{1}} ^{{n}} \:\frac{\mathrm{1}}{{k}} \\ $$

Question Number 63164 Answers: 0 Comments: 0

let S_n =Σ_(k=1) ^n (((−1)^k )/k) and H_n =Σ_(k=1) ^n (1/k) calculate S_n interms of H_n 2)find lim_(n→+∞) S_n

$${let}\:{S}_{{n}} =\sum_{{k}=\mathrm{1}} ^{{n}} \:\:\frac{\left(−\mathrm{1}\right)^{{k}} }{{k}}\:\:\:\:\:\:{and}\:{H}_{{n}} =\sum_{{k}=\mathrm{1}} ^{{n}} \:\frac{\mathrm{1}}{{k}} \\ $$$${calculate}\:{S}_{{n}} \:{interms}\:{of}\:{H}_{{n}} \\ $$$$\left.\mathrm{2}\right){find}\:{lim}_{{n}\rightarrow+\infty} \:{S}_{{n}} \\ $$

Question Number 63162 Answers: 1 Comments: 3

Find the set of values of x which satisfy the inequalities (2/(x−1))≤(1/x) and x^2 −∣3x∣+2<0

$${Find}\:{the}\:{set}\:{of}\:{values}\:{of}\:{x}\:{which}\:{satisfy}\:{the}\:{inequalities}\: \\ $$$$\frac{\mathrm{2}}{{x}−\mathrm{1}}\leqslant\frac{\mathrm{1}}{{x}}\:\:{and}\:\:{x}^{\mathrm{2}} −\mid\mathrm{3}{x}\mid+\mathrm{2}<\mathrm{0} \\ $$

Question Number 63154 Answers: 0 Comments: 5

Given that α and β are the roots oc the equation ax^2 +bx+c=0 . Show that λμb^2 = ac(λ + μ)^2 , where (α/β)= (λ/μ).

$${Given}\:{that}\:\alpha\:{and}\:\beta\:{are}\:{the}\:{roots}\:{oc}\:{the}\:{equation}\:{ax}^{\mathrm{2}} +{bx}+{c}=\mathrm{0} \\ $$$$.\:{Show}\:{that}\:\:\:\lambda\mu{b}^{\mathrm{2}} =\:{ac}\left(\lambda\:+\:\mu\right)^{\mathrm{2}} ,\:{where}\:\frac{\alpha}{\beta}=\:\frac{\lambda}{\mu}. \\ $$

Question Number 63152 Answers: 0 Comments: 0

The Variables x and y satisfy the differential equation (d^2 y/dx^2 )−x(dy/dx) + y = x^2 use the approximations ((d^2 y/dx^2 ))_n ≈ ((y_(n+1) −2y_n +y_(n−1) )/h^(2 ) ) and ((dy/dx))≈((y_(n+1) −y_(n−1) )/(2h )) to show that (2−hx_n )y_(n+1) ≈2x_n ^2 + 398y_n −(200 + 10x_n )y_(n−1) given that y=1 and x=0 and that y_(−1) =y_1 show that y_1 =0.995 Evaluate y when x=0.3, giving your answercorrect to 3 decimal places

$${The}\:{Variables}\:{x}\:{and}\:{y}\:{satisfy}\:{the}\:{differential}\:{equation}\: \\ $$$$\:\frac{{d}^{\mathrm{2}} {y}}{{dx}^{\mathrm{2}} }−{x}\frac{{dy}}{{dx}}\:+\:{y}\:=\:{x}^{\mathrm{2}} \:\:{use}\:{the}\:{approximations} \\ $$$$\:\:\:\left(\frac{{d}^{\mathrm{2}} {y}}{{dx}^{\mathrm{2}} }\right)_{{n}} \approx\:\frac{{y}_{{n}+\mathrm{1}} −\mathrm{2}{y}_{{n}} +{y}_{{n}−\mathrm{1}} }{{h}^{\mathrm{2}\:} }\:{and}\:\:\left(\frac{{dy}}{{dx}}\right)\approx\frac{{y}_{{n}+\mathrm{1}} −{y}_{{n}−\mathrm{1}} }{\mathrm{2}{h}\:}\:{to}\:{show}\:{that} \\ $$$$\:\:\left(\mathrm{2}−{hx}_{{n}} \right){y}_{{n}+\mathrm{1}} \approx\mathrm{2}{x}_{{n}} ^{\mathrm{2}} \:+\:\mathrm{398}{y}_{{n}} −\left(\mathrm{200}\:+\:\mathrm{10}{x}_{{n}} \right){y}_{{n}−\mathrm{1}} \\ $$$${given}\:{that}\:{y}=\mathrm{1}\:{and}\:{x}=\mathrm{0}\:{and}\:{that}\:{y}_{−\mathrm{1}} ={y}_{\mathrm{1}} \:{show}\:{that}\:{y}_{\mathrm{1}} =\mathrm{0}.\mathrm{995} \\ $$$${Evaluate}\:{y}\:{when}\:{x}=\mathrm{0}.\mathrm{3},\:{giving}\:{your}\:{answercorrect}\:{to}\:\mathrm{3}\:{decimal}\:{places} \\ $$

Question Number 63143 Answers: 0 Comments: 0

Question Number 63139 Answers: 0 Comments: 3

Σ_(n = 1) ^m ((log n)/n^(3/2) )

$$\underset{\mathrm{n}\:=\:\mathrm{1}} {\overset{\mathrm{m}} {\sum}}\:\frac{\mathrm{log}\:\mathrm{n}}{\mathrm{n}^{\mathrm{3}/\mathrm{2}} } \\ $$

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