1-x-5-1-dx-

Question Number 194975 by youssef last updated on 21/Jul/23

$$\int\frac{\mathrm{1}}{\boldsymbol{{x}}^{\mathrm{5}} +\mathrm{1}}\boldsymbol{{dx}} \\ $$

Commented by Frix last updated on 21/Jul/23

You have to decompose x^5 +1=(x+1)(x^2 −((1+(√5))/2)x+1)(x^2 −((1−(√5))/2)x+1)

$$\mathrm{You}\:\mathrm{have}\:\mathrm{to}\:\mathrm{decompose} \\ $$$${x}^{\mathrm{5}} +\mathrm{1}=\left({x}+\mathrm{1}\right)\left({x}^{\mathrm{2}} −\frac{\mathrm{1}+\sqrt{\mathrm{5}}}{\mathrm{2}}{x}+\mathrm{1}\right)\left({x}^{\mathrm{2}} −\frac{\mathrm{1}−\sqrt{\mathrm{5}}}{\mathrm{2}}{x}+\mathrm{1}\right) \\ $$

Commented by York12 last updated on 22/Jul/23

$${how}\:{do}\:{you}\:{know}\:{how}\:{to}\:{decompose}\:{it} \\ $$

Commented by TheHoneyCat last updated on 22/Jul/23

Let D(X)=Π_(k=1) ^n (X−α_k )^d_k With all the α_k distinct and (obviously) of total degree d_D =Σ_(k=1) ^n d_k And let N(X) be of degree d_N <d_D Then ∃!β_(k,i) : (N/D)=Σ_(k=1) ^n Σ_(i=1) ^d_k (β_(k,i) /((X−α_k )^i )) Note that if you don′t want to use C you can google for an alternate version where you allow for fractions of the form X^2 +γX+δ to replace the (X−α) To find the β coeficients, you can simply just eyball the thing (as they are unique) a few smart techiques are to look at well chosen values and limits of (N/D) and also, feel free to try some of them separately (for instance it is sometimes ussfull to obtain all the β_(k,i) for a given k before going further).

$$\mathrm{Let}\: \\ $$$${D}\left({X}\right)=\underset{{k}=\mathrm{1}} {\overset{{n}} {\prod}}\left({X}−\alpha_{{k}} \right)^{{d}_{{k}} } \\ $$$$\mathrm{With}\:\mathrm{all}\:\mathrm{the}\:\alpha_{{k}} \:\mathrm{distinct}\:\mathrm{and}\:\left(\mathrm{obviously}\right)\:\mathrm{of}\: \\ $$$$\mathrm{total}\:\mathrm{degree}\:{d}_{{D}} =\underset{{k}=\mathrm{1}} {\overset{{n}} {\sum}}{d}_{{k}} \\ $$$$\mathrm{And}\:\mathrm{let}\:{N}\left({X}\right)\:\mathrm{be}\:\mathrm{of}\:\mathrm{degree}\:{d}_{{N}} <{d}_{{D}} \\ $$$$\mathrm{Then}\:\exists!\beta_{{k},{i}} : \\ $$$$\frac{{N}}{{D}}=\underset{{k}=\mathrm{1}} {\overset{{n}} {\sum}}\underset{{i}=\mathrm{1}} {\overset{{d}_{{k}} } {\sum}}\frac{\beta_{{k},{i}} }{\left({X}−\alpha_{{k}} \right)^{{i}} } \\ $$$$ \\ $$$$\mathrm{Note}\:\mathrm{that}\:\mathrm{if}\:\mathrm{you}\:\mathrm{don}'\mathrm{t}\:\mathrm{want}\:\mathrm{to}\:\mathrm{use}\:\mathbb{C}\:\mathrm{you}\:\mathrm{can} \\ $$$$\mathrm{google}\:\mathrm{for}\:\mathrm{an}\:\mathrm{alternate}\:\mathrm{version}\:\mathrm{where}\:\mathrm{you} \\ $$$$\mathrm{allow}\:\mathrm{for}\:\mathrm{fractions}\:\mathrm{of}\:\mathrm{the}\:\mathrm{form}\:{X}^{\mathrm{2}} +\gamma{X}+\delta \\ $$$$\mathrm{to}\:\mathrm{replace}\:\mathrm{the}\:\left({X}−\alpha\right) \\ $$$$ \\ $$$$\mathrm{To}\:\mathrm{find}\:\mathrm{the}\:\beta\:\mathrm{coeficients},\:\mathrm{you}\:\mathrm{can}\:\mathrm{simply}\:\mathrm{just} \\ $$$$\mathrm{eyball}\:\mathrm{the}\:\mathrm{thing}\:\left(\mathrm{as}\:\mathrm{they}\:\mathrm{are}\:\mathrm{unique}\right) \\ $$$$\mathrm{a}\:\mathrm{few}\:\mathrm{smart}\:\mathrm{techiques}\:\mathrm{are}\:\mathrm{to}\:\mathrm{look}\:\mathrm{at}\:\mathrm{well}\:\mathrm{chosen} \\ $$$$\mathrm{values}\:\mathrm{and}\:\mathrm{limits}\:\mathrm{of}\:\:\frac{{N}}{{D}} \\ $$$$\mathrm{and}\:\mathrm{also},\:\mathrm{feel}\:\mathrm{free}\:\mathrm{to}\:\mathrm{try}\:\mathrm{some}\:\mathrm{of}\:\mathrm{them}\:\mathrm{separately} \\ $$$$\left(\mathrm{for}\:\mathrm{instance}\:\mathrm{it}\:\mathrm{is}\:\mathrm{sometimes}\:\mathrm{ussfull}\:\mathrm{to}\:\mathrm{obtain}\right. \\ $$$$\left.\mathrm{all}\:\mathrm{the}\:\beta_{{k},{i}} \:\mathrm{for}\:\mathrm{a}\:\mathrm{given}\:{k}\:\mathrm{before}\:\mathrm{going}\:\mathrm{further}\right). \\ $$

Commented by York12 last updated on 22/Jul/23

$${thanks}\:{sir}\: \\ $$$${and}\:{what}\:{books}\:{do}\:{you}\:{recommend} \\ $$$${for}\:{algebra} \\ $$$$ \\ $$$$ \\ $$

Commented by TheHoneyCat last updated on 22/Jul/23

I don't know, I learned at school workout books. I you know french, BibMath is a good website (but general)

Commented by York12 last updated on 22/Jul/23

$${okay}\:{sir}\:{thanks} \\ $$$$ \\ $$

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