求广义积分∫dx/（xx+x-2）（积分号上面为正无穷,下面为2）

Question

求广义积分 ∫dx/（xx+x-2） （积分号上面为正无穷,下面为2）

Answer 1

∫(2->+∞) dx/(x²+x-2)
= ∫(2->+∞) dx/[(x+2)(x-1)]
= ∫(2->+∞) {1/[3(x+2)] - 1/[3(x-1)]} dx
= (1/3)∫(2->+∞) [ln|x+2| - ln|x-1|] + C
= (1/3)lim(M->+∞) ln[(x+2)/(x-1)] - (1/3)lim(N->2) ln[(x+2)/(x-1)]
= (1/3)lim(M->+∞) ln[(1+2/x)/(1-1/x)] - (1/3)*ln[(2+2)/(2-1)]
= (1/3)*ln[(1+0)/(1-0)] - (1/3)ln4
= (1/3)*ln1 - (1/3)*2ln2
= (2ln2)/3