c_0+c_1/(x - 1) + (c_2 x + c_3)/(x + 2)^2 - x^3/((x - 1)(x + 2)^2)=0c0+c1x−1+c2x+c3(x+2)2−x3(x−1)(x+2)2=0
Solving for c_0,c_1,c_2,c_3c0,c1,c2,c3 we have
c_0=2,c_1=2/9,c_2=-56/9,c_3=-64/9c0=2,c1=29,c2=−569,c3=−649 and then
int x^3/((x - 1)(x + 2)^2)dx=int(c_0+c_1/(x - 1) + (c_2 x + c_3)/(x + 2)^2)dx=∫x3(x−1)(x+2)2dx=∫(c0+c1x−1+c2x+c3(x+2)2)dx=
=c_0 x + (2 c_2 - c_3)/(x+2) + c_1 Log(x-1) + c_2 Log(x+2)+C=c0x+2c2−c3x+2+c1log(x−1)+c2log(x+2)+C