分部积分法：∫f(x)dg(x)=f(x)g(x)-∫g(x)df(x)
先做个变量置换,令x^1/2=t
则∫ln(1+x^1/2)dx=∫ln(1+t)d(t^2)
=ln(1+t)(t^2)-∫(t^2)dln(1+t)
=ln(1+t)(t^2)-∫(t^2)/(1+t)dt
=ln(1+t)(t^2)-∫(t^2-1)/(1+t)dt+∫1/(1+t)dt
=ln(1+t)(t^2)-∫(t-1)dt+∫1/(1+t)dt
=ln(1+t)(t^2)-1/2(t^2)+t-ln(1+t)
=xln(1+x^1/2)-1/2x+(x^1/2)-ln(1+x^1/2)