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Question Number 57700 Answers: 1 Comments: 0

R(1 − cosθ) = 0.5 Rsinθ = 4 R = ? θ = ?

$$\mathrm{R}\left(\mathrm{1}\:−\:\mathrm{cos}\theta\right)\:=\:\mathrm{0}.\mathrm{5} \\ $$$$\mathrm{Rsin}\theta\:=\:\mathrm{4} \\ $$$$\mathrm{R}\:=\:? \\ $$$$\theta\:=\:? \\ $$

Question Number 57698 Answers: 1 Comments: 1

Question Number 57695 Answers: 0 Comments: 0

Question Number 57688 Answers: 1 Comments: 12

Solve for n: Σ_i ^(n − 1) ^n C_i 2^i = 65, n ∈ Z^+ . where zero is included

$$\:\:\mathrm{Solve}\:\mathrm{for}\:\:\mathrm{n}:\:\:\:\:\:\:\:\:\underset{\mathrm{i}} {\overset{\mathrm{n}\:−\:\mathrm{1}} {\sum}}\:\:\:\overset{\mathrm{n}} {\:}\mathrm{C}_{\mathrm{i}} \:\mathrm{2}^{\mathrm{i}} \:\:=\:\:\mathrm{65},\:\:\:\:\:\:\:\:\:\:\:\:\mathrm{n}\:\in\:\mathbb{Z}^{+} .\:\:\:\:\mathrm{where}\:\:\mathrm{zero}\:\mathrm{is}\: \\ $$$$\:\:\mathrm{included} \\ $$

Question Number 57669 Answers: 1 Comments: 2

Question Number 57668 Answers: 0 Comments: 3

let V_n = ∫_0 ^(1+(1/n)) ((x+1)/(√(2x^2 +3))) dx 1) calculate lim_(n→+∞) V_n 2) find nature of the serie Σ V_n

$${let}\:{V}_{{n}} =\:\int_{\mathrm{0}} ^{\mathrm{1}+\frac{\mathrm{1}}{{n}}} \:\:\:\:\frac{{x}+\mathrm{1}}{\sqrt{\mathrm{2}{x}^{\mathrm{2}} \:+\mathrm{3}}}\:{dx}\:\: \\ $$$$\left.\mathrm{1}\right)\:{calculate}\:{lim}_{{n}\rightarrow+\infty} \:{V}_{{n}} \\ $$$$\left.\mathrm{2}\right)\:{find}\:{nature}\:{of}\:{the}\:{serie}\:\Sigma\:{V}_{{n}} \\ $$

Question Number 57667 Answers: 0 Comments: 3

calculate U_n =∫_(π/n) ^((2π)/n) (dx/(2 +sinx)) 1) calculate U_n and lim_(n→+∞) nU_n 2) find nature of Σ U_n

$${calculate}\:{U}_{{n}} =\int_{\frac{\pi}{{n}}} ^{\frac{\mathrm{2}\pi}{{n}}} \:\:\:\:\:\frac{{dx}}{\mathrm{2}\:+{sinx}} \\ $$$$\left.\mathrm{1}\right)\:{calculate}\:{U}_{{n}} \:\:\:\:\:\:\:{and}\:{lim}_{{n}\rightarrow+\infty} \:\:{nU}_{{n}} \\ $$$$\left.\mathrm{2}\right)\:{find}\:{nature}\:{of}\:\Sigma\:{U}_{{n}} \\ $$

Question Number 57666 Answers: 0 Comments: 3

1) calculate f(θ) =∫_0 ^1 (√(t^2 +2sinθt +1))dt with 0≤θ≤(π/2) 2) calculate g(t) =∫_0 ^1 (√(t^2 +2(sinθ)t +1))dθ 3) find also h(θ) =∫_0 ^1 (t/(√(t^2 +2(sinθ)t +1)))dt

$$\left.\mathrm{1}\right)\:{calculate}\:{f}\left(\theta\right)\:=\int_{\mathrm{0}} ^{\mathrm{1}} \sqrt{{t}^{\mathrm{2}} \:+\mathrm{2}{sin}\theta{t}\:+\mathrm{1}}{dt}\:\:\:\:{with}\:\mathrm{0}\leqslant\theta\leqslant\frac{\pi}{\mathrm{2}} \\ $$$$\left.\mathrm{2}\right)\:{calculate}\:{g}\left({t}\right)\:=\int_{\mathrm{0}} ^{\mathrm{1}} \sqrt{{t}^{\mathrm{2}} \:+\mathrm{2}\left({sin}\theta\right){t}\:+\mathrm{1}}{d}\theta \\ $$$$\left.\mathrm{3}\right)\:{find}\:{also}\:{h}\left(\theta\right)\:=\int_{\mathrm{0}} ^{\mathrm{1}} \:\:\:\:\frac{{t}}{\sqrt{{t}^{\mathrm{2}} \:+\mathrm{2}\left({sin}\theta\right){t}\:+\mathrm{1}}}{dt} \\ $$

Question Number 57665 Answers: 0 Comments: 4

let f(a) =∫_(π/4) ^(π/3) (√(a+tan^2 x))dx with a>0 1) find a explicit form of f(a) 2) find also g(a) =∫_(π/4) ^(π/3) (dx/(√(a+tan^2 x))) 3) find the values of ∫_(π/4) ^(π/3) (√(2+tan^2 x))dx and ∫_(π/4) ^(π/3) (dx/(√(3+tan^2 x)))

$${let}\:{f}\left({a}\right)\:=\int_{\frac{\pi}{\mathrm{4}}} ^{\frac{\pi}{\mathrm{3}}} \sqrt{{a}+{tan}^{\mathrm{2}} {x}}{dx}\:\:\:{with}\:{a}>\mathrm{0} \\ $$$$\left.\mathrm{1}\right)\:{find}\:{a}\:{explicit}\:{form}\:{of}\:{f}\left({a}\right) \\ $$$$\left.\mathrm{2}\right)\:\:{find}\:{also}\:{g}\left({a}\right)\:=\int_{\frac{\pi}{\mathrm{4}}} ^{\frac{\pi}{\mathrm{3}}} \:\:\frac{{dx}}{\sqrt{{a}+{tan}^{\mathrm{2}} {x}}} \\ $$$$\left.\mathrm{3}\right)\:{find}\:{the}\:{values}\:{of}\:\:\int_{\frac{\pi}{\mathrm{4}}} ^{\frac{\pi}{\mathrm{3}}} \:\sqrt{\mathrm{2}+{tan}^{\mathrm{2}} {x}}{dx}\:\:{and}\:\int_{\frac{\pi}{\mathrm{4}}} ^{\frac{\pi}{\mathrm{3}}} \:\:\:\frac{{dx}}{\sqrt{\mathrm{3}+{tan}^{\mathrm{2}} {x}}} \\ $$

Question Number 57664 Answers: 0 Comments: 1

A luminous point P is inside a circle. A ray emanates from P and after two reflections by the circle,returns to P. If θ be the angle of incidence, a= the distance of P from the centre of the circle and b=the distance of the centre from the point where the ray in its course crosses its diameter through P. prove that tan θ=((a−b)/(a+b))

$${A}\:{luminous}\:{point}\:{P}\:\:{is}\:{inside}\:{a}\:{circle}. \\ $$$${A}\:{ray}\:{emanates}\:{from}\:{P}\:{and}\:{after}\:{two} \\ $$$${reflections}\:{by}\:{the}\:{circle},{returns}\:{to}\:{P}. \\ $$$${If}\:\theta\:{be}\:{the}\:{angle}\:{of}\:{incidence},\:{a}=\:{the} \\ $$$${distance}\:{of}\:{P}\:{from}\:{the}\:{centre}\:{of}\:{the} \\ $$$${circle}\:{and}\:{b}={the}\:{distance}\:{of}\:{the}\:{centre} \\ $$$${from}\:{the}\:{point}\:{where}\:{the}\:{ray}\:{in}\:{its} \\ $$$${course}\:{crosses}\:{its}\:{diameter}\:{through}\:{P}. \\ $$$$ \\ $$$${prove}\:{that}\:\mathrm{tan}\:\theta=\frac{{a}−{b}}{{a}+{b}} \\ $$

Question Number 57653 Answers: 0 Comments: 5

is it possible to find the exact value of I? I=∫_0 ^π sin (sin x) dx

$$\mathrm{is}\:\mathrm{it}\:\mathrm{possible}\:\mathrm{to}\:\mathrm{find}\:\mathrm{the}\:\mathrm{exact}\:\mathrm{value}\:\mathrm{of}\:{I}? \\ $$$${I}=\underset{\mathrm{0}} {\overset{\pi} {\int}}\mathrm{sin}\:\left(\mathrm{sin}\:{x}\right)\:{dx} \\ $$

Question Number 57647 Answers: 0 Comments: 2

find approximate value of ξ(3) by using n−1 ≤n≤n+1 for n integr natural .

$${find}\:\:{approximate}\:{value}\:{of}\:\xi\left(\mathrm{3}\right)\:{by}\:{using}\:\:\:{n}−\mathrm{1}\:\leqslant{n}\leqslant{n}+\mathrm{1}\:\:\:{for}\:{n}\:{integr} \\ $$$${natural}\:. \\ $$

Question Number 57635 Answers: 1 Comments: 2

Find the sum of the cubes of first n even number, and first n odd number.

$$\mathrm{Find}\:\mathrm{the}\:\mathrm{sum}\:\mathrm{of}\:\mathrm{the}\:\mathrm{cubes}\:\mathrm{of}\:\mathrm{first}\:\mathrm{n}\:\mathrm{even}\:\mathrm{number},\:\:\mathrm{and} \\ $$$$\mathrm{first}\:\mathrm{n}\:\mathrm{odd}\:\mathrm{number}. \\ $$

Question Number 57633 Answers: 1 Comments: 0