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Question Number 42906    Answers: 3   Comments: 0

x^x +y^y =31 x+y = 5 Find x and y

$${x}^{{x}} +{y}^{{y}} =\mathrm{31} \\ $$$${x}+{y}\:=\:\mathrm{5} \\ $$$${Find}\:{x}\:{and}\:{y} \\ $$

Question Number 42897    Answers: 2   Comments: 1

Question Number 42895    Answers: 1   Comments: 1

Question Number 42878    Answers: 1   Comments: 0

Question Number 42870    Answers: 0   Comments: 0

let 0<x<1 and Γ(x) =∫_0 ^∞ t^(x−1) e^(−t) dt 1) prove that Γ(x).Γ(1−x) =(π/(sin(πx))) (compliments formulae) 2) calculate Γ(n) and Γ(n+(1/2)) with n from N.

$${let}\:\mathrm{0}<{x}<\mathrm{1}\:\:{and}\:\Gamma\left({x}\right)\:=\int_{\mathrm{0}} ^{\infty} \:{t}^{{x}−\mathrm{1}} \:{e}^{−{t}} \:{dt}\: \\ $$$$\left.\mathrm{1}\right)\:{prove}\:{that}\:\Gamma\left({x}\right).\Gamma\left(\mathrm{1}−{x}\right)\:=\frac{\pi}{{sin}\left(\pi{x}\right)}\:\:\:\left({compliments}\:{formulae}\right) \\ $$$$\left.\mathrm{2}\right)\:{calculate}\:\Gamma\left({n}\right)\:{and}\:\Gamma\left({n}+\frac{\mathrm{1}}{\mathrm{2}}\right)\:{with}\:{n}\:{from}\:{N}. \\ $$

Question Number 42866    Answers: 2   Comments: 0

Question Number 42863    Answers: 2   Comments: 7

Question Number 42861    Answers: 1   Comments: 0

14÷2(3+4)=?

$$\mathrm{14}\boldsymbol{\div}\mathrm{2}\left(\mathrm{3}+\mathrm{4}\right)=? \\ $$

Question Number 42826    Answers: 0   Comments: 7

solving ax^4 +bx^3 +cx^2 +dx+e=0 (a≠0, b, c, d, e)∈Q special cases (easy to solve) ax^4 +e=0 solve at^2 +e=0 ⇒ x=±(√t_(1, 2) ) ax^4 +cx^2 +e=0 solve at^2 +ct+e=0 ⇒ x=±(√t_(1, 2) ) always try all factors of ±e because a(x−α)(x−β)(x−γ)(x−δ)=ax^4 +...+αβγδ ⇒ e=αβγδ next we must find the nature of the solutions 4 real solutions 2 real & 2 complex solutions 4 complex solutions a, b, c, d, e ∈Q ⇒ complex solutions always in conjugated pairs draw the function or calculate some values to find the number of real solutions divide by a x^4 +px^3 +qx^2 +rx+s=0 [p=(b/a) q=(c/a) r=(d/a) s=(e/a)] I′ll soon post some cases I′ve been able to solve as comments

$$\mathrm{solving} \\ $$$${ax}^{\mathrm{4}} +{bx}^{\mathrm{3}} +{cx}^{\mathrm{2}} +{dx}+{e}=\mathrm{0} \\ $$$$\left({a}\neq\mathrm{0},\:{b},\:{c},\:{d},\:{e}\right)\in\mathbb{Q} \\ $$$$ \\ $$$$\:\:\:\:\:\:\:\:\:\:\mathrm{special}\:\mathrm{cases}\:\left(\mathrm{easy}\:\mathrm{to}\:\mathrm{solve}\right) \\ $$$$\:\:\:\:\:\:\:\:\:\:{ax}^{\mathrm{4}} +{e}=\mathrm{0}\:\mathrm{solve}\:{at}^{\mathrm{2}} +{e}=\mathrm{0}\:\Rightarrow\:{x}=\pm\sqrt{{t}_{\mathrm{1},\:\mathrm{2}} } \\ $$$$\:\:\:\:\:\:\:\:\:\:{ax}^{\mathrm{4}} +{cx}^{\mathrm{2}} +{e}=\mathrm{0}\:\mathrm{solve}\:{at}^{\mathrm{2}} +{ct}+{e}=\mathrm{0}\:\Rightarrow\:{x}=\pm\sqrt{{t}_{\mathrm{1},\:\mathrm{2}} } \\ $$$$ \\ $$$$\mathrm{always}\:\mathrm{try}\:\mathrm{all}\:\mathrm{factors}\:\mathrm{of}\:\pm{e} \\ $$$$\mathrm{because}\:{a}\left({x}−\alpha\right)\left({x}−\beta\right)\left({x}−\gamma\right)\left({x}−\delta\right)={ax}^{\mathrm{4}} +...+\alpha\beta\gamma\delta \\ $$$$\Rightarrow\:{e}=\alpha\beta\gamma\delta \\ $$$$ \\ $$$$\mathrm{next}\:\mathrm{we}\:\mathrm{must}\:\mathrm{find}\:\mathrm{the}\:\mathrm{nature}\:\mathrm{of}\:\mathrm{the}\:\mathrm{solutions} \\ $$$$\mathrm{4}\:\mathrm{real}\:\mathrm{solutions} \\ $$$$\mathrm{2}\:\mathrm{real}\:\&\:\mathrm{2}\:\mathrm{complex}\:\mathrm{solutions} \\ $$$$\mathrm{4}\:\mathrm{complex}\:\mathrm{solutions} \\ $$$${a},\:{b},\:{c},\:{d},\:{e}\:\in\mathbb{Q}\:\Rightarrow\:\mathrm{complex}\:\mathrm{solutions}\:\mathrm{always}\:\mathrm{in} \\ $$$$\mathrm{conjugated}\:\mathrm{pairs} \\ $$$$\mathrm{draw}\:\mathrm{the}\:\mathrm{function}\:\mathrm{or}\:\mathrm{calculate}\:\mathrm{some}\:\mathrm{values} \\ $$$$\mathrm{to}\:\mathrm{find}\:\mathrm{the}\:\mathrm{number}\:\mathrm{of}\:\mathrm{real}\:\mathrm{solutions} \\ $$$$ \\ $$$$\mathrm{divide}\:\mathrm{by}\:{a} \\ $$$${x}^{\mathrm{4}} +{px}^{\mathrm{3}} +{qx}^{\mathrm{2}} +{rx}+{s}=\mathrm{0} \\ $$$$\left[{p}=\frac{{b}}{{a}}\:\:{q}=\frac{{c}}{{a}}\:\:{r}=\frac{{d}}{{a}}\:\:{s}=\frac{{e}}{{a}}\right] \\ $$$$ \\ $$$$\mathrm{I}'\mathrm{ll}\:\mathrm{soon}\:\mathrm{post}\:\mathrm{some}\:\mathrm{cases}\:\mathrm{I}'\mathrm{ve}\:\mathrm{been}\:\mathrm{able}\:\mathrm{to}\:\mathrm{solve} \\ $$$$\mathrm{as}\:\mathrm{comments} \\ $$

Question Number 42823    Answers: 1   Comments: 1

Evaluate : ∫_(−5) ^( 5) x^2 [x+(1/2)]dx = ? where [.]= greatest integer function

$$\mathrm{Evaluate}\:: \\ $$$$\int_{−\mathrm{5}} ^{\:\mathrm{5}} \:{x}^{\mathrm{2}} \left[{x}+\frac{\mathrm{1}}{\mathrm{2}}\right]{dx}\:=\:\:? \\ $$$${where}\:\left[.\right]=\:{greatest}\:{integer}\:{function} \\ $$

Question Number 42812    Answers: 0   Comments: 1

study the convervence of ∫_1 ^(+∞) ((arctan(x−1))/((x^2 −1)^(4/3) )) dx

$${study}\:{the}\:{convervence}\:{of}\:\:\int_{\mathrm{1}} ^{+\infty} \:\:\frac{{arctan}\left({x}−\mathrm{1}\right)}{\left({x}^{\mathrm{2}} −\mathrm{1}\right)^{\frac{\mathrm{4}}{\mathrm{3}}} }\:{dx} \\ $$

Question Number 42810    Answers: 1   Comments: 1

find ∫_0 ^∞ (x^5 /(1+x^7 ))dx .

$${find}\:\:\int_{\mathrm{0}} ^{\infty} \:\:\:\:\frac{{x}^{\mathrm{5}} }{\mathrm{1}+{x}^{\mathrm{7}} }{dx}\:\:. \\ $$

Question Number 42809    Answers: 1   Comments: 1

calculate ∫_0 ^∞ ((tdt)/((1+t^4 )^2 ))

$${calculate}\:\int_{\mathrm{0}} ^{\infty} \:\:\:\frac{{tdt}}{\left(\mathrm{1}+{t}^{\mathrm{4}} \right)^{\mathrm{2}} } \\ $$

Question Number 42808    Answers: 1   Comments: 0

let f(x) = ∫_x ^(2x) ((1+cos(t))/(√(t^4 −t^2 +4)))dt 1) find D_f 2) calculate f^′ (x) 3) find lim_(x→+∞) f(x)

$${let}\:{f}\left({x}\right)\:=\:\int_{{x}} ^{\mathrm{2}{x}} \:\:\frac{\mathrm{1}+{cos}\left({t}\right)}{\sqrt{{t}^{\mathrm{4}} −{t}^{\mathrm{2}} \:+\mathrm{4}}}{dt} \\ $$$$\left.\mathrm{1}\right)\:{find}\:{D}_{{f}} \\ $$$$\left.\mathrm{2}\right)\:{calculate}\:\:{f}^{'} \left({x}\right) \\ $$$$\left.\mathrm{3}\right)\:{find}\:{lim}_{{x}\rightarrow+\infty} \:{f}\left({x}\right) \\ $$

Question Number 42806    Answers: 0   Comments: 0

let u_n = ∫_n ^(n+2) (((t+n)^(1/4) )/t^(1/3) )dt find lim_(n→+∞) u_n

$${let}\:\:{u}_{{n}} =\:\int_{{n}} ^{{n}+\mathrm{2}} \:\:\:\frac{\left({t}+{n}\right)^{\frac{\mathrm{1}}{\mathrm{4}}} }{{t}^{\frac{\mathrm{1}}{\mathrm{3}}} }{dt} \\ $$$${find}\:{lim}_{{n}\rightarrow+\infty} \:{u}_{{n}} \\ $$

Question Number 42805    Answers: 1   Comments: 1

calculate lim_(n→+∞) S_n with S_n = (1/n^4 ) Σ_(k=1) ^n (k^3 /(√((1+((k/n))^2 )^3 )))

$${calculate}\:{lim}_{{n}\rightarrow+\infty} \:{S}_{{n}} \:\:\:{with} \\ $$$${S}_{{n}} =\:\frac{\mathrm{1}}{{n}^{\mathrm{4}} }\:\sum_{{k}=\mathrm{1}} ^{{n}} \:\:\frac{{k}^{\mathrm{3}} }{\sqrt{\left(\mathrm{1}+\left(\frac{{k}}{{n}}\right)^{\mathrm{2}} \right)^{\mathrm{3}} }} \\ $$

Question Number 42804    Answers: 1   Comments: 1

calculate ∫_(1/2) ^1 (dx/((√(4x^2 −1)) +(√(4x^2 +1))))

$${calculate}\:\:\int_{\frac{\mathrm{1}}{\mathrm{2}}} ^{\mathrm{1}} \:\:\:\:\:\frac{{dx}}{\sqrt{\mathrm{4}{x}^{\mathrm{2}} \:−\mathrm{1}}\:+\sqrt{\mathrm{4}{x}^{\mathrm{2}} \:+\mathrm{1}}} \\ $$

Question Number 42803    Answers: 1   Comments: 0

find ∫_0 ^1 (x^2 +1)(√((1−x)/(1+x)))dx

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

Question Number 42802    Answers: 0   Comments: 1

calculate ∫_(1/2) ^(5/4) (x^3 /(√(2+x−x^2 )))dx

$${calculate}\:\:\int_{\frac{\mathrm{1}}{\mathrm{2}}} ^{\frac{\mathrm{5}}{\mathrm{4}}} \:\:\:\frac{{x}^{\mathrm{3}} }{\sqrt{\mathrm{2}+{x}−{x}^{\mathrm{2}} }}{dx} \\ $$

Question Number 42801    Answers: 1   Comments: 1

find f(x) = ∫_(π/4) ^(π/3) ((cosxdx)/(2cos^2 x +sin^2 x +1))

$${find}\:{f}\left({x}\right)\:=\:\int_{\frac{\pi}{\mathrm{4}}} ^{\frac{\pi}{\mathrm{3}}} \:\:\:\:\frac{{cosxdx}}{\mathrm{2}{cos}^{\mathrm{2}} {x}\:+{sin}^{\mathrm{2}} {x}\:+\mathrm{1}} \\ $$

Question Number 42800    Answers: 1   Comments: 0

find ∫ ((sinx)/(1+2 cosx))dx

$${find}\:\int\:\:\:\:\:\frac{{sinx}}{\mathrm{1}+\mathrm{2}\:{cosx}}{dx} \\ $$

Question Number 42799    Answers: 0   Comments: 2

let I = ∫_0 ^(π/8) e^(−2t) cos^4 t and J =∫_0 ^(π/8) e^(−2t) sin^4 dt find the values of I andJ .

$${let}\:{I}\:=\:\int_{\mathrm{0}} ^{\frac{\pi}{\mathrm{8}}} \:\:{e}^{−\mathrm{2}{t}} \:{cos}^{\mathrm{4}} {t}\:\:\:\:{and}\:{J}\:\:=\int_{\mathrm{0}} ^{\frac{\pi}{\mathrm{8}}} \:{e}^{−\mathrm{2}{t}} \:{sin}^{\mathrm{4}} {dt} \\ $$$${find}\:{the}\:{values}\:{of}\:{I}\:{andJ}\:. \\ $$

Question Number 42798    Answers: 1   Comments: 2

find I_n = ∫_0 ^1 x^n (√(1−x^2 ))dx

$${find}\:{I}_{{n}} =\:\int_{\mathrm{0}} ^{\mathrm{1}} \:{x}^{{n}} \sqrt{\mathrm{1}−{x}^{\mathrm{2}} }{dx} \\ $$

Question Number 42797    Answers: 0   Comments: 1

let u_k = ∫_(−(π/2) +kπ) ^(−(π/2) +(k+1)π) e^(−t) cost dt 1) calculate u_k 2) let A_n =Σ_(k=0) ^n u_k find lim_(n→+∞) A_n

$${let}\:{u}_{{k}} =\:\int_{−\frac{\pi}{\mathrm{2}}\:+{k}\pi} ^{−\frac{\pi}{\mathrm{2}}\:+\left({k}+\mathrm{1}\right)\pi} \:\:{e}^{−{t}} \:{cost}\:{dt} \\ $$$$\left.\mathrm{1}\right)\:{calculate}\:{u}_{{k}} \\ $$$$\left.\mathrm{2}\right)\:{let}\:{A}_{{n}} =\sum_{{k}=\mathrm{0}} ^{{n}} \:{u}_{{k}} \:\:\:\:\:{find}\:{lim}_{{n}\rightarrow+\infty} \:{A}_{{n}} \\ $$

Question Number 42796    Answers: 1   Comments: 1

calculate I = ∫_0 ^1 (x^2 /(1+x^2 )) arctan(x)dx

$${calculate}\:{I}\:=\:\int_{\mathrm{0}} ^{\mathrm{1}} \:\:\frac{{x}^{\mathrm{2}} }{\mathrm{1}+{x}^{\mathrm{2}} }\:{arctan}\left({x}\right){dx} \\ $$

Question Number 42795    Answers: 0   Comments: 3

calculate ∫_(−∞) ^(+∞) (dx/((1+x^2 +x^4 )^2 ))

$${calculate}\:\:\int_{−\infty} ^{+\infty} \:\:\:\:\frac{{dx}}{\left(\mathrm{1}+{x}^{\mathrm{2}} \:+{x}^{\mathrm{4}} \right)^{\mathrm{2}} } \\ $$

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