Add and subtract 44 to the numerator:
int x^2/(4-x^2)dx = -int (x^2-4+4)/(x^2-4)dx ∫x24−x2dx=−∫x2−4+4x2−4dx
int x^2/(4-x^2)dx = -int( 1+4/(x^2-4))dx ∫x24−x2dx=−∫(1+4x2−4)dx
using the linearity of the integral:
int x^2/(4-x^2)dx = -int dx - 4int dx/(x^2-4)∫x24−x2dx=−∫dx−4∫dxx2−4
int x^2/(4-x^2)dx = -x - 4int dx/(x^2-4)∫x24−x2dx=−x−4∫dxx2−4
Decompose now the resulting integrand using partial fractions:
4/(x^2-4) = 4/((x-2)(x+2))4x2−4=4(x−2)(x+2)
4/(x^2-4) = A/(x-2)+B/(x+2)4x2−4=Ax−2+Bx+2
4/(x^2-4) = (A(x+2)+B(x-2))/((x-2)(x+2))4x2−4=A(x+2)+B(x−2)(x−2)(x+2)
4 = Ax +2A +Bx -2B4=Ax+2A+Bx−2B
4 = (A+B)x +2(A-B)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