∫0到2 ln(x+√(x^2+1))dx怎么求?
问题描述：
∫0到2 ln(x+√(x^2+1))dx怎么求?
答
用分步积
∫[0,2] ln(x+√(x^2+1))dx
=x ln(x+√(x^2+1))[0,2]-∫[0,2] xdln(x+√(x^2+1))
=2ln(2+√5)-∫[0,2] x/(x+√(x^2+1))*(1+x/√(x^2+1)dx
=2ln(2+√5)-∫[0,2] x/√(x^2+1)dx
=2ln(2+√5)-∫[0,2] 1/2*1/√(x^2+1)dx^2
=2ln(2+√5)-1/3(x^2+1)^(3/2)[0,2]
=2ln(2+√5)-9+1/3