Question

∫(x+1)/x^2-2x+2dx等于什么

Answer 1

∫ (x+1)/(x²-2x+2) dx

= ∫ [(1/2)(2x-2)+2]/(x²-2x+2) dx

= (1/2)∫ (2x-2)/(x²-2x+2) dx + ∫ 2/(x²-2x+2) dx

= (1/2)∫ d(x²-2x+2)/(x²-2x+2) + 2∫ 1/[(x-1)²+1] d(x-1)

= (1/2)ln| x²-2x+2 | + 2arctan(x-1) + C

Note：

Let x+1 = A(2x-2)+B

x+1 = 2Ax + (B-2A)

2A=1 => A=1/2

B-2A=1 => B=1+2(1/2)=2

∴x+1 = (1/2)(2x-2)+2