Question

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

Answer 1

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.]