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Question Number 57198 Answers: 0 Comments: 2

Question Number 57175 Answers: 0 Comments: 5

Question Number 57174 Answers: 1 Comments: 0

Question Number 57173 Answers: 0 Comments: 0

Question Number 57164 Answers: 1 Comments: 0

2x+1+x^2 −x^3 +x^4 −x^5 −.....=((13)/6) solved equation. ∣x∣<1

$$\mathrm{2}\boldsymbol{\mathrm{x}}+\mathrm{1}+\boldsymbol{\mathrm{x}}^{\mathrm{2}} −\boldsymbol{\mathrm{x}}^{\mathrm{3}} +\boldsymbol{\mathrm{x}}^{\mathrm{4}} −\boldsymbol{\mathrm{x}}^{\mathrm{5}} −.....=\frac{\mathrm{13}}{\mathrm{6}} \\ $$$$\boldsymbol{\mathrm{solved}}\:\:\boldsymbol{\mathrm{equation}}. \\ $$$$\mid\boldsymbol{\mathrm{x}}\mid<\mathrm{1} \\ $$

Question Number 57163 Answers: 0 Comments: 0

Question Number 57140 Answers: 1 Comments: 0

∫_( 0) ^1 (√((1+x)(1+x^3 ))) dx ≤ ((15)/8)

$$\underset{\:\mathrm{0}} {\overset{\mathrm{1}} {\int}}\:\sqrt{\left(\mathrm{1}+{x}\right)\left(\mathrm{1}+{x}^{\mathrm{3}} \right)}\:{dx}\:\leqslant\:\frac{\mathrm{15}}{\mathrm{8}} \\ $$

Question Number 57138 Answers: 1 Comments: 1

The value of the integral ∫_( 0) ^π (1/(a^2 −2a cos x+1)) dx (a >1) is

$$\mathrm{The}\:\mathrm{value}\:\mathrm{of}\:\mathrm{the}\:\mathrm{integral} \\ $$$$\underset{\:\mathrm{0}} {\overset{\pi} {\int}}\:\:\:\frac{\mathrm{1}}{{a}^{\mathrm{2}} −\mathrm{2}{a}\:\mathrm{cos}\:{x}+\mathrm{1}}\:{dx}\:\:\left({a}\:>\mathrm{1}\right)\:\:\mathrm{is} \\ $$

Question Number 57136 Answers: 1 Comments: 0

∫_a ^b ((f(x))/(f(x)+f(a+b−x))) dx =

$$\underset{{a}} {\overset{{b}} {\int}}\:\:\frac{{f}\left({x}\right)}{{f}\left({x}\right)+{f}\left({a}+{b}−{x}\right)}\:{dx}\:= \\ $$

Question Number 57127 Answers: 0 Comments: 24

{cos1°}+{cos2°}+{cos3°}+....+{cos270}=?

$$\left\{\boldsymbol{\mathrm{cos}}\mathrm{1}°\right\}+\left\{\boldsymbol{\mathrm{cos}}\mathrm{2}°\right\}+\left\{\boldsymbol{\mathrm{cos}}\mathrm{3}°\right\}+....+\left\{\boldsymbol{\mathrm{cos}}\mathrm{270}\right\}=? \\ $$

Question Number 57124 Answers: 1 Comments: 3

Question Number 59160 Answers: 0 Comments: 0

let f(x) =∫_0 ^∞ ((cos(xcosθ))/(x^2 +θ^2 )) dθ and g(x) =∫_0 ^∞ ((sin(xcosθ))/(x^2 +θ^2 )) dθ 1) find a explicit form of f(x) and g(x) 2) find the value of ∫_0 ^∞ ((cos(2cosθ))/(4+θ^2 )) dθ and ∫_0 ^∞ ((sin(2cosθ))/(4+θ^2 )) dθ 3) let u_n =f(n^2 ) study the serie Σ u_n

$${let}\:{f}\left({x}\right)\:=\int_{\mathrm{0}} ^{\infty} \:\:\frac{{cos}\left({xcos}\theta\right)}{{x}^{\mathrm{2}} \:+\theta^{\mathrm{2}} }\:{d}\theta\:\:\:\:\:\:{and}\:\:{g}\left({x}\right)\:=\int_{\mathrm{0}} ^{\infty} \:\:\:\:\:\frac{{sin}\left({xcos}\theta\right)}{{x}^{\mathrm{2}} \:+\theta^{\mathrm{2}} }\:{d}\theta \\ $$$$\left.\mathrm{1}\right)\:{find}\:{a}\:{explicit}\:{form}\:{of}\:{f}\left({x}\right)\:{and}\:{g}\left({x}\right) \\ $$$$\left.\mathrm{2}\right)\:{find}\:{the}\:{value}\:{of}\:\int_{\mathrm{0}} ^{\infty} \:\:\:\frac{{cos}\left(\mathrm{2}{cos}\theta\right)}{\mathrm{4}+\theta^{\mathrm{2}} }\:{d}\theta\:\:{and}\:\:\int_{\mathrm{0}} ^{\infty} \:\:\:\frac{{sin}\left(\mathrm{2}{cos}\theta\right)}{\mathrm{4}+\theta^{\mathrm{2}} }\:{d}\theta \\ $$$$\left.\mathrm{3}\right)\:{let}\:{u}_{{n}} ={f}\left({n}^{\mathrm{2}} \right)\:\:\:{study}\:\:{the}\:{serie}\:\Sigma\:{u}_{{n}} \\ $$

Question Number 57115 Answers: 0 Comments: 0

Question Number 57114 Answers: 1 Comments: 4

Question Number 57103 Answers: 0 Comments: 2

let A_n =∫∫_w_n e^(−x^2 −y^2 ) (√(x^2 +y^2 ))dxdy with w_n =[(1/n),n]×[(1/n),n] 1) calculate A_n interms of n 2) find lim_(n→+∞) A_n

$${let}\:{A}_{{n}} =\int\int_{{w}_{{n}} } \:{e}^{−{x}^{\mathrm{2}} −{y}^{\mathrm{2}} } \:\sqrt{{x}^{\mathrm{2}} \:+{y}^{\mathrm{2}} }{dxdy} \\ $$$${with}\:{w}_{{n}} =\left[\frac{\mathrm{1}}{{n}},{n}\right]×\left[\frac{\mathrm{1}}{{n}},{n}\right] \\ $$$$\left.\mathrm{1}\right)\:{calculate}\:{A}_{{n}} {interms}\:{of}\:{n} \\ $$$$\left.\mathrm{2}\right)\:{find}\:{lim}_{{n}\rightarrow+\infty} \:{A}_{{n}} \\ $$

Question Number 57097 Answers: 0 Comments: 2

Question Number 57084 Answers: 4 Comments: 2

Question Number 57075 Answers: 1 Comments: 5

Question Number 57074 Answers: 1 Comments: 0

Solve the system. ^x C_(y + 1) = 20, ^(x − 1) C_y = 10

$$\mathrm{Solve}\:\mathrm{the}\:\mathrm{system}. \\ $$$$\:\:\:\:\:\:\overset{\mathrm{x}} {\:}\mathrm{C}_{\mathrm{y}\:+\:\mathrm{1}} \:\:=\:\:\mathrm{20},\:\:\:\:\:\:\:\:\:\:\:\overset{\mathrm{x}\:−\:\mathrm{1}} {\:}\mathrm{C}_{\mathrm{y}} \:\:=\:\:\mathrm{10} \\ $$

Question Number 57073 Answers: 0 Comments: 2

f(x)=A f : [0, ∞) g : [1, 0] g(x)=B lim_(n→∞) (1/n)∫_0 ^n f(x)g((x/n))dx=...

$${f}\left({x}\right)={A} \\ $$$${f}\::\:\left[\mathrm{0},\:\infty\right)\: \\ $$$${g}\::\:\left[\mathrm{1},\:\mathrm{0}\right] \\ $$$${g}\left({x}\right)={B} \\ $$$$\underset{{n}\rightarrow\infty} {\mathrm{lim}}\:\frac{\mathrm{1}}{{n}}\int_{\mathrm{0}} ^{{n}} {f}\left({x}\right){g}\left(\frac{{x}}{{n}}\right){dx}=... \\ $$

Question Number 57071 Answers: 1 Comments: 1

Question Number 57062 Answers: 2 Comments: 2

find the sum of all three digital natural numbers that are divisible by 7

$${find}\:{the}\:{sum}\:{of}\:{all} \\ $$$${three}\:{digital}\:{natural} \\ $$$${numbers}\:{that}\:{are}\: \\ $$$${divisible}\:{by}\:\mathrm{7} \\ $$

Question Number 57060 Answers: 0 Comments: 0

Question Number 57051 Answers: 0 Comments: 5

F_n = ((ϕ^n − (1 − ϕ)^n )/(√5)) , with { ((ϕ = ((1 + (√5))/2))) :} (G_n ) : { ((G_1 = 1)),((G_2 = 1)),((G_m = G_(m−1) + G_(m−2) )) :} Give a proof for F_n = G_n , ∀n∈N^∗ . Thank you

$$ \\ $$$$\:\:\boldsymbol{{F}}_{\boldsymbol{{n}}} \:=\:\frac{\varphi^{{n}} \:−\:\left(\mathrm{1}\:−\:\varphi\right)^{{n}} }{\sqrt{\mathrm{5}}}\:,\:\:\mathrm{with}\:\begin{cases}{\varphi\:=\:\frac{\mathrm{1}\:+\:\sqrt{\mathrm{5}}}{\mathrm{2}}}\end{cases} \\ $$$$ \\ $$$$\:\:\left(\boldsymbol{{G}}_{\boldsymbol{{n}}} \right)\::\:\begin{cases}{\boldsymbol{{G}}_{\mathrm{1}} \:=\:\mathrm{1}}\\{\boldsymbol{{G}}_{\mathrm{2}} \:=\:\mathrm{1}}\\{\boldsymbol{{G}}_{{m}} \:=\:\boldsymbol{{G}}_{\mathrm{m}−\mathrm{1}} \:+\:\boldsymbol{{G}}_{{m}−\mathrm{2}} }\end{cases} \\ $$$$ \\ $$$$\:\:\mathrm{Give}\:\mathrm{a}\:\mathrm{proof}\:\mathrm{for}\:\boldsymbol{{F}}_{\boldsymbol{{n}}} \:=\:\boldsymbol{{G}}_{\boldsymbol{{n}}} \:,\:\forall{n}\in\mathbb{N}^{\ast} .\: \\ $$$$\:\:\mathrm{Thank}\:\mathrm{you} \\ $$

Question Number 57048 Answers: 2 Comments: 1

Question Number 57046 Answers: 0 Comments: 1

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