Question Number 104772 by mathmax by abdo last updated on 23/Jul/20

let ϕ(x) = x^3  +x+1  1) prove that ϕ have one real root α  2)determine a approximate value for α  by use of newton method  3)factorise inside R(x) f(x)  4) calculste ∫ (dx/(ϕ(x)))

Answered by MAB last updated on 23/Jul/20

1) ϕ′(x)=3x^2 +1>0   lim_(x→−∞) ϕ(x)=−∞  lim_(x→+∞) ϕ(x)=+∞  hence ϕ is a bijection of ]−∞,+∞[ to  itself, ϕ has a unique real root  2)x_(n+1) =x_n −((ϕ(x_n ))/(ϕ′(x_n )))  x_(n+1) =x_n −((x_n ^3 +x_n +1)/(3x_n ^2 +1))  x_(n+1) =((2x_n ^3 −1)/(3x_n ^2 −1))  let x_0 =0  using python x_5 =−0.6823278039465127  to be continued...

Commented byabdomsup last updated on 23/Jul/20

thank you sir.

Commented byMAB last updated on 23/Jul/20

you are welcome sir

Answered by MAB last updated on 23/Jul/20

3) ϕ(x)=(x−α)(x^2 +αx+1+α^2 )  (easy to check)  4)∫(dx/(ϕ(x)))=∫((1/((α^2 +1)))((1/(x−α))−(x/(x^2 +αx+α^2 +1)))dx  =(1/(α^2 +1))(ln(x−α)−(1/2)ln(x^2 +αx+α^2 +1)+2α((arctan(((α+2x)/(√(2α^2 +4)))))/(√(3α^2 +4))))+C