计算积分∫(5z-2）/z(z-1)^2dz

Question

计算积分∫(5z-2）/z(z-1)^2dz

Answer 1

∫ (5z-2)/[z(z-1)²] dz
= ∫ [5(z-1+1)-2]/[z(z-1)²] dz
= ∫ [5(z-1)+3]/[z(z-1)²] dz
= 5∫ dz/[z(z-1)]+3∫ dz/[z(z-1)²]
= -5∫ (z-1-z)/[z(z-1)] dz+3∫ (z+1-z)/[z(z-1)²] dz
= -5∫ [1/z-1/(z-1)] dz+3∫ {1/(z-1)²-1/[z(z-1)]} dz
= -5[ln|z|-ln|z-1|] + 3∫ d(z-1)/(z-1)²+3∫ (z-1-z)/[z(z-1)] dz
= -5ln|z|+5ln|z-1|-3/(z-1)+3∫ [1/z-1/(z-1)] dz
= -5ln|z|+5ln|z-1|-3/(z-1)+3ln|z|-3ln|z-1|+C
= -2ln|z|+2ln|z-1|-3/(z-1)+C
= 2ln|(z-1)/z|-3/(z-1)+C