how-I-calculate-1-x-8-x-2-dx-

Question Number 76053 by john santuy last updated on 23/Dec/19

h o w I c a l c u l a t e \int \frac{1}{x^{8} + x^{2}} d x ?

Commented by abdomathmax last updated on 23/Dec/19

d e c o m p o s e t h e f r a c t i o n F (x) = \frac{1}{x^{2} + x^{8}} a n d s e e

t h a t F (x) = \frac{1}{x^{2} (x^{6} + 1)} = \frac{1}{x^{2} ({(x^{2})}^{3} + 1)}

= \frac{1}{x^{2} (x^{2} + 1) (x^{4} - x^{2} + 1)} = \dots .

Commented by benjo last updated on 23/Dec/19

thanks sir

Answered by MJS last updated on 23/Dec/19

\frac{1}{x^{8} + x^{2}} = \frac{1}{x^{2} (x^{6} + 1)} =

= \frac{1}{x^{2}} - \frac{1}{3 (x^{2} + 1)} + \frac{\sqrt{3} x + 1}{6 (x^{2} + \sqrt{3} x + 1)} - \frac{\sqrt{3} x - 1}{6 (x^{2} - \sqrt{3} x + 1)}

now solve these

Commented by benjo last updated on 23/Dec/19

thanks sir]

Commented by vishalbhardwaj last updated on 23/Dec/19

sir this procedure is very difficult to undersatand

, please explain this once again when you are direct

ly taken any value of x

Commented by MJS last updated on 23/Dec/19

the decomposing or the integration ?

Commented by vishalbhardwaj last updated on 23/Dec/19

decomposing

Commented by MJS last updated on 24/Dec/19

x^{8} + x^{2} = x^{2} (x^{6} + 1) =

= x^{2} ({(x^{2})}^{3} + 1) =

[\begin{matrix} p^{3} + 1 = (p + 1) (p^{2} - p + 1) \end{matrix}]

= x^{2} (x^{2} + 1) (x^{4} - x^{2} + 1) =

[\begin{matrix} x^{4} - x^{2} + 1 = 0 let x = \sqrt{q} \\ q^{2} - q + 1 = 0 \Rightarrow q = - \frac{1}{2} \pm \frac{\sqrt{3}}{2} i \\ these are the 2^{nd} and 3^{rd} solutions \\ of q^{3} + 1 = 0 \Rightarrow q = e^{\pm i \frac{π}{3}} \\ \Rightarrow x = e^{\pm i \frac{π}{6}} \lor x = e^{\pm i \frac{5 π}{6}} \\ \Rightarrow x^{4} - x^{2} + 1 = \\ = (x - e^{- i \frac{π}{6}}) (x - e^{i \frac{π}{6}}) (x - e^{- i \frac{5 π}{6}}) (x - e^{i \frac{5 π}{6}}) = \\ = (x^{2} - \sqrt{3} x + 1) (x^{2} + \sqrt{3} x + 1) \end{matrix}]

= x^{2} (x^{2} + 1) (x^{2} - \sqrt{3} x + 1) (x^{2} + \sqrt{3} x + 1)

\Rightarrow

\frac{1}{x^{8} + x^{2}} = \frac{a x + b}{x^{2}} + \frac{c x + d}{x^{2} + 1} + \frac{e x + f}{x^{2} - \sqrt{3} x + 1} + \frac{g x + h}{x^{2} + \sqrt{3} x + 1}

now use your (hopefully) learned method

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