What is the definite integral of #ln x# from 0 to 1?

1 Answer
Sep 2, 2015

The integral should give you #-1#.

Explanation:

We have:
#int_0^1ln(x)dx=# by parts: #=xlnx-intx1/xdx==xlnx-int1dx=xlnx-x|_0^1#
#=[1ln(1)-1]-[0ln(0)-0]=#
But #ln(0)# cannot be evaluated:
#=[0-1]-[?]#
Considering the meaning of integral (the area described by a curve and the #x# axis) and the graph of your function:
graph{ln(x) [-2.375, 17.625, -8.44, 1.56]}
you can see that your function at zero continues indefinitely towards #-oo# giving you a never ending area!

BUT
#ln(0)# may not exist, but does #lim_(x→0+)xln(x)# exist? I think you will find it is #0# and hence the value of the definite integral is #−1#.
[After @George C.]