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AllQuestion and Answers: Page 1496

Question Number 57785 Answers: 1 Comments: 2

Question Number 57784 Answers: 0 Comments: 0

Question Number 57783 Answers: 1 Comments: 0

Question Number 57779 Answers: 0 Comments: 0

kno_3 ⇒k_2 o+n_2 +o_2

$${kno}_{\mathrm{3}} \Rightarrow{k}_{\mathrm{2}} {o}+{n}_{\mathrm{2}} +{o}_{\mathrm{2}} \\ $$

Question Number 57770 Answers: 1 Comments: 0

Question Number 57754 Answers: 2 Comments: 1

f(x)=ln(x) (f○f)′=?

$$\boldsymbol{{f}}\left(\boldsymbol{{x}}\right)=\boldsymbol{{ln}}\left(\boldsymbol{{x}}\right) \\ $$$$\left(\boldsymbol{{f}}\circ\boldsymbol{{f}}\right)'=? \\ $$

Question Number 57750 Answers: 1 Comments: 0

find ∫ x^2 (√(25−x^2 ))dx

$${find}\:\int\:{x}^{\mathrm{2}} \sqrt{\mathrm{25}−{x}^{\mathrm{2}} }{dx}\: \\ $$

Question Number 57749 Answers: 1 Comments: 3

find ∫ (dx/(x^2 (√(9+x^2 ))))

$${find}\:\int\:\:\frac{{dx}}{{x}^{\mathrm{2}} \sqrt{\mathrm{9}+{x}^{\mathrm{2}} }} \\ $$

Question Number 57748 Answers: 2 Comments: 2

find ∫ x^2 (√(4+x^2 ))dx

$${find}\:\int\:{x}^{\mathrm{2}} \sqrt{\mathrm{4}+{x}^{\mathrm{2}} }{dx} \\ $$

Question Number 57746 Answers: 0 Comments: 4

let f(x)=∫_(−∞) ^(+∞) (dt/((t^2 −2xt +1)^2 )) with ∣x∣<1 (x real) 1) determine a explicit form for f(x) 2) find also g(x) =∫_(−∞) ^(+∞) ((tdt)/((t^2 −2xt +1)^3 )) 3) calculate ∫_(−∞) ^(+∞) (dt/((t^2 −(√2)t +1)^2 )) and ∫_(−∞) ^(+∞) ((tdt)/((t^2 −(√2)t +1)^3 )) 4) calculate A(θ) =∫_(−∞) ^(+∞) (dt/((t^2 −2cosθ t+1)^2 )) and B(θ) =∫_(−∞) ^(+∞) ((tdt)/((t^2 −2cosθ t +1)^3 )) with 0<θ <(π/2) .

$${let}\:{f}\left({x}\right)=\int_{−\infty} ^{+\infty} \:\:\:\:\frac{{dt}}{\left({t}^{\mathrm{2}} −\mathrm{2}{xt}\:+\mathrm{1}\right)^{\mathrm{2}} }\:\:{with}\:\mid{x}\mid<\mathrm{1}\:\:\:\left({x}\:{real}\right) \\ $$$$\left.\mathrm{1}\right)\:{determine}\:{a}\:{explicit}\:{form}\:\:{for}\:{f}\left({x}\right) \\ $$$$\left.\mathrm{2}\right)\:{find}\:{also}\:{g}\left({x}\right)\:=\int_{−\infty} ^{+\infty} \:\:\:\frac{{tdt}}{\left({t}^{\mathrm{2}} −\mathrm{2}{xt}\:+\mathrm{1}\right)^{\mathrm{3}} } \\ $$$$\left.\mathrm{3}\right)\:{calculate}\:\int_{−\infty} ^{+\infty} \:\:\:\frac{{dt}}{\left({t}^{\mathrm{2}} −\sqrt{\mathrm{2}}{t}\:+\mathrm{1}\right)^{\mathrm{2}} }\:\:\:{and}\:\int_{−\infty} ^{+\infty} \:\:\frac{{tdt}}{\left({t}^{\mathrm{2}} −\sqrt{\mathrm{2}}{t}\:+\mathrm{1}\right)^{\mathrm{3}} } \\ $$$$\left.\mathrm{4}\right)\:{calculate}\:{A}\left(\theta\right)\:=\int_{−\infty} ^{+\infty} \:\:\:\frac{{dt}}{\left({t}^{\mathrm{2}} \:−\mathrm{2}{cos}\theta\:{t}+\mathrm{1}\right)^{\mathrm{2}} }\:\:\:{and}\: \\ $$$${B}\left(\theta\right)\:=\int_{−\infty} ^{+\infty} \:\:\:\:\frac{{tdt}}{\left({t}^{\mathrm{2}} \:−\mathrm{2}{cos}\theta\:{t}\:+\mathrm{1}\right)^{\mathrm{3}} }\:\:\:\:{with}\:\mathrm{0}<\theta\:<\frac{\pi}{\mathrm{2}}\:\:\:\:\:. \\ $$

Question Number 57736 Answers: 1 Comments: 0

Can we use L′Ho^ pital′s rule if we have a fraction in the form (+∞)/(−∞) ?

$${Can}\:{we}\:{use}\:{L}'{H}\hat {{o}pital}'{s}\:{rule}\:{if}\:{we}\:{have} \\ $$$${a}\:{fraction}\:{in}\:{the}\:{form}\:\left(+\infty\right)/\left(−\infty\right)\:\:\:? \\ $$

Question Number 57735 Answers: 2 Comments: 0

Question Number 57719 Answers: 1 Comments: 0

Find all complex number z that satisfy sinh z = i

$$\mathrm{Find}\:\mathrm{all}\:\mathrm{complex}\:\mathrm{number}\:{z}\:\mathrm{that}\:\mathrm{satisfy} \\ $$$$\mathrm{sinh}\:{z}\:=\:{i} \\ $$

Question Number 57706 Answers: 0 Comments: 0

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 .

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

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