设x1>x2>-1
f(x1)-f(x2)=ln(x1+1)-ln(x2+1)
=ln(x1+1)/(x2+1)
=ln[(x2+1)+(x1-x2)]/(x2+1)
=ln[1+(x1-x2)/(x2+1)]
>ln1=0
即
f(x1)>f(x2)
所以由增函数的定义,知
f(X)在定义域上为增函数.