求由2到5的定积分∫1/1+√x-1dx,

Question

求由2到5的定积分 ∫1/1+√x-1dx,

Answer 1

∫[2-->5] 1/(1+√(x-1))dx
令√(x-1)=u,则x=u²+1,dx=2udu,u：1-->2
=∫[1-->2] 2u/(1+u)du
=2∫[1-->2] (u+1-1)/(1+u)du
=2∫[1-->2] 1du-2∫[1-->2] 1/(1+u)du
=2u-2ln(1+u) [1-->2]
=2-2ln3+2ln2
=2-2ln(3/2)
或2+2ln(2/3)