计算∫(4,9)][√x/(√x-1)]dx

Question

计算∫(4,9)][√x/(√x-1)]dx

Answer 1

计算[4,9)]∫(√x)/[(√x)-1)]dx
令√x=u,则x=u²,dx=2udu,x=4时u=2,x=9时u=3,于是：
原式=[2,3]2∫[u²/(u-1)]du=[2,3]2∫[(u+1)+1/(u-1)]du=[2,3]2[∫(u+1)d(u+1)+∫d(u-1)/(u-1)]
=2[(u+1)²/2+ln(u-1)]︱[2,3]=2[8+ln2-9/2]=7+2ln2.