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

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Question Number 33589 by abdo imad last updated on 19/Apr/18
1) decompose F(x) = (1/((x^2 +4)(x−3)^2 )) 2) calculate ∫_4 ^(+∞) (dx/((x^2 +4)(x−3)^2 )) .
1)decomposeF(x)=1(x2+4)(x−3)22)calculate∫4+∞dx(x2+4)(x−3)2.
Commented by math khazana by abdo last updated on 22/Apr/18
F(x)= (a/(x−3)) +(b/((x−3)^2 )) +((cx+d)/(x^2 +4)) b =lim_(x→3) (x−3)^2 F(x)= (1/(13)) lim_(x→+∞) xF(x)=0 =a+c ⇒c=−a ⇒ F(x)= (a/(x−3)) +(1/(13(x−3)^2 )) +((−ax +d)/(x^2 +4)) F(0) = (1/(36)) = ((−a)/3) + (1/(117)) +(d/4) ⇒1=−12a +((36)/(117)) +9d ⇒ −12a +9d = 1−((36)/(117)) = ((81)/(117)) F(2)= (1/8) = −a +(1/(13)) +((−2a+d)/8) ⇒ 1= −8a +(8/(13)) −2a +d =−10a +d +(8/(13)) ⇒ −10a +d = 1−(8/(13)) = (5/(13)) we get tbe system −12a +9d = ((81)/(117)) and −10a +d =(5/(13)) ⇒ −90a +9d = ((45)/(13)) ⇒ 78a = ((81)/(117)) −((45)/(13)) ⇒ a= (1/(78))( ((81)/(117)) −((45)/(13))).....be continued....
F(x)=ax−3+b(x−3)2+cx+dx2+4b=limx→3(x−3)2F(x)=113limx→+∞xF(x)=0=a+c⇒c=−a⇒F(x)=ax−3+113(x−3)2+−ax+dx2+4F(0)=136=−a3+1117+d4⇒1=−12a+36117+9d⇒−12a+9d=1−36117=81117F(2)=18=−a+113+−2a+d8⇒1=−8a+813−2a+d=−10a+d+813⇒−10a+d=1−813=513wegettbesystem−12a+9d=81117and−10a+d=513⇒−90a+9d=4513⇒78a=81117−4513⇒a=178(81117−4513)…..becontinued….

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