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Question Number 33589 by abdo imad last updated on 19/Apr/18

$$1)decomposeF\left(x\right)=\frac{1}{(x^2+4){(x-3)}^2}$$$$2)calculate{\int }_4^{+\mathrm{\infty }}\frac{dx}{(x^2+4){(x-3)}^2}.$$

Commented by math khazana by abdo last updated on 22/Apr/18

$$F\left(x\right)=\frac{a}{x-3}+\frac{b}{{(x-3)}^2}+\frac{cx+d}{x^2+4}$$$$b={lim}_{x\to 3}{(x-3)}^{2}F\left(x\right)=\frac{1}{13}$$$${lim}_{x\to +\mathrm{\infty }}xF\left(x\right)=0=a+c\Rightarrow c=-a\Rightarrow$$$$F\left(x\right)=\frac{a}{x-3}+\frac{1}{13{(x-3)}^2}+\frac{-ax+d}{x^2+4}$$$$F\left(0\right)=\frac{1}{36}=\frac{-a}{3}+\frac{1}{117}+\frac{d}{4}\Rightarrow 1=-12a+\frac{36}{117}+9d$$$$\Rightarrow -12a+9d=1-\frac{36}{117}=\frac{81}{117}$$$$F\left(2\right)=\frac{1}{8}=-a+\frac{1}{13}+\frac{-2a+d}{8}\Rightarrow$$$$1=-8a+\frac{8}{13}-2a+d=-10a+d+\frac{8}{13}\Rightarrow$$$$-10a+d=1-\frac{8}{13}=\frac{5}{13}wegettbesystem$$$$-12a+9d=\frac{81}{117}and-10a+d=\frac{5}{13}\Rightarrow$$$$-90a+9d=\frac{45}{13}\Rightarrow 78a=\frac{81}{117}-\frac{45}{13}\Rightarrow$$$$a=\frac{1}{78}(\frac{81}{117}-\frac{45}{13}).....becontinued....$$

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