1-decompose-F-x-1-x-2-4-3-x-2-1-2-2-find-3-dx-x-2-4-3-x-2-1-2-

Question Number 130029 by mathmax by abdo last updated on 22/Jan/21

1) decompose F (x) = \frac{1}{{(x^{2} - 4)}^{3} {(x^{2} + 1)}^{2}}

2) find \int_{3}^{\infty} \frac{dx}{{(x^{2} - 4)}^{3} {(x^{2} + 1)}^{2}}

Commented by MJS_new last updated on 22/Jan/21

Ostrogradski gives

F (x) = - \frac{x (17 x^{4} - 331 x^{2} + 1252)}{16000 {(x^{2} - 4)}^{2} (x^{2} + 1)} - \frac{1}{16000} \int \frac{17 x^{2} - 687}{(x - 2) (x + 2) (x^{2} + 1)} d x =

= - \frac{x (17 x^{4} - 331 x^{2} + 1252)}{16000 {(x^{2} - 4)}^{2} (x^{2} + 1)} + \frac{619}{320000} \int \frac{1}{x + 2} - \frac{1}{x - 2} d x - \frac{11}{1250} \int \frac{d x}{x^{2} + 1} =

= - \frac{x (17 x^{4} - 331 x^{2} + 1252)}{16000 {(x^{2} - 4)}^{2} (x^{2} + 1)} + \frac{619}{320000} \ln ∣ \frac{x - 2}{x + 2} ∣ - \frac{11}{1250} \arctan x + C

\Rightarrow answer is \frac{619}{320000} \ln 5 - \frac{11}{1250} \arctan \frac{1}{3} - \frac{21}{80000}

Commented by liberty last updated on 22/Jan/21

your favorite method sir hahaha

Answered by Ar Brandon last updated on 22/Jan/21

f (a, b)

f (a, b) = \frac{1}{(x^{2} - a^{2}) (x^{2} + b^{2})} = \frac{α}{x - a} + \frac{β}{x + a} + \frac{λ x + μ}{x^{2} + b^{2}}

= \frac{α (x + a) (x^{2} + b^{2}) + β (x - a) (x^{2} + b^{2}) + (λ x + μ) (x^{2} - a^{2})}{(x^{2} - a^{2}) (x^{2} + b^{2})}

α = \frac{1}{2 a (a^{2} + b^{2})}, β = \frac{1}{- 2 a (a^{2} + b^{2})}

α {ab}^{2} - β {ab}^{2} - μ a^{2} = 1 \Rightarrow μ a^{2} = (\frac{\cancel {a b}^{2}}{2 \cancel a (a^{2} + b^{2})}) + (\frac{\cancel {a b}^{2}}{2 \cancel a (a^{2} + b^{2})}) - 1

\Rightarrow μ = - \frac{1}{(a^{2} + b^{2})}, α + β + λ = 0 \Rightarrow λ = 0

f (a, b) = \frac{1}{2 a (a^{2} + b^{2})} (\frac{1}{x - a} - \frac{1}{x + a}) - \frac{1}{(a^{2} + b^{2})} \cdot \frac{1}{x^{2} + b^{2}}

\int f (a, b) dx = \frac{\ln ∣ x - a ∣ - \ln ∣ x + a ∣}{2 a (a^{2} + b^{2})} - \frac{\tan^{- 1} (b / x)}{b (a^{2} + b^{2})} + C

F (x) = \int {\frac{\partial^{3} f (a, b)}{\partial a^{3}} \cdot \frac{\partial^{2} f (a, b)}{\partial b^{2}}} dx