Question Number 104505 by mathmax by abdo last updated on 22/Jul/20

calculate ∫_0 ^∞   (dx/((x+1)^2 (x+2)^2 (x+3)^2 ))

Answered by OlafThorendsen last updated on 22/Jul/20

R(x) =  (1/((x+1)^2 (x+2)^2 (x+3)^2 ))  R(x) =  (1/(4(x+1)^2 ))+(1/((x+2)^2 ))+(1/(4(x+3)^2 ))  −(3/(4(x+1)))+(3/(4(x+3)))  ∫_0 ^∞  R(x)dx =  [−(1/(4(x+1)))−(1/(x+2))−(1/(4(x+3)))−(3/4)ln∣((x+1)/(x+3))∣]_0 ^∞   = (1/4)+(1/2)+(1/(12))+(3/4)ln(1/3)  = (5/6)−(3/4)ln3