x^2/(x^2-1)=x^2/[(x+1)(x-1)]
令x^2/[(x+1)(x-1)]=A+B/(x+1)+C/(x-1)
x^2=A(x^2-1)+B(x-1)+C(x+1)
解得A=1,B=-1/2,C=1/2
∫x^2/(x^2-1) dx
=∫dx-(1/2)∫dx/(x+1)+(1/2)∫dx/(x-1)
=x-(1/2)|x+1|+(1/2)ln|x-1|+C
=x+(1/2)ln|(x-1)/(x+1)|+C