求有理函数积分：①∫x^4/(x-1)^3dx ②∫1/x^2(1+2x)dx ③∫x/x^3-x^2+x-1dx.
问题描述：
求有理函数积分：①∫x^4/(x-1)^3dx ②∫1/x^2(1+2x)dx ③∫x/x^3-x^2+x-1dx.
答
∫x^4dx/(x-1)^3= -(1/2)∫x^4d(x-1)^(-2)=(-1/2)x^4/(x-1)^2+(1/2)∫4x^3dx/(x-1)^2=(-1/2)x^4/(x-1)^2+(-2)∫x^3d(x-1)^(-1)=(-1/2)x^4/(x-1)^2-2x^3/(x-1)+2∫3x^2dx/(x-1)=(-1/2)x^4/(x-1)^2-2x^3/(x-1) +6∫(x+...