Add and subtract #4# to the numerator:
#int x^2/(4-x^2)dx = -int (x^2-4+4)/(x^2-4)dx #
#int x^2/(4-x^2)dx = -int( 1+4/(x^2-4))dx #
using the linearity of the integral:
#int x^2/(4-x^2)dx = -int dx - 4int dx/(x^2-4)#
#int x^2/(4-x^2)dx = -x - 4int dx/(x^2-4)#
Decompose now the resulting integrand using partial fractions:
#4/(x^2-4) = 4/((x-2)(x+2))#
#4/(x^2-4) = A/(x-2)+B/(x+2)#
#4/(x^2-4) = (A(x+2)+B(x-2))/((x-2)(x+2))#
#4 = Ax +2A +Bx -2B#
#4 = (A+B)x +2(A-B)#
#{(A+B=0),(A-B = 2):}#
#{(A=1),(B=-1):}#
So:
#int x^2/(4-x^2)dx = -x - int dx/(x-2) + int dx/(x+2)#
#int x^2/(4-x^2)dx = -x - ln abs (x-2) + ln abs (x+2) +C#
