Question

x2+5x+2 / (x+1)(x2+1) 的部分分式和

Answer 1

设(x^2+5x+2)/[(x+1)(x^2+1)]

=a/(x+1)+(bx+c)/(x^2+1),

去分母得x^2+5x+2=a(x^2+1)+(bx+c)(x+1)

=(a+b)x^2+(b+c)x+a+c,

比较得a+b=1,b+c=5,a+c=2,

解得a=-1,b=2,c=3.

∴(x^2+5x+2)/[(x+1)(x^2+1)]

=-1/(x+1)+(2x+3)/(x^2+1).