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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 ))