What is the antiderivative of #ln x#?

1 Answer
Jun 29, 2016

#intlnxdx=xlnx-x+C#

Explanation:

The integral (antiderivative) of #lnx# is an interesting one, because the process to find it is not what you'd expect.

We will be using integration by parts to find #intlnxdx#:
#intudv=uv-intvdu#
Where #u# and #v# are functions of #x#.

Here, we let:
#u=lnx->(du)/dx=1/x->du=1/xdx# and #dv=dx->intdv=intdx->v=x#

Making necessary substitutions into the integration by parts formula, we have:
#intlnxdx=(lnx)(x)-int(x)(1/xdx)#
#->(lnx)(x)-intcancel(x)(1/cancelxdx)#
#=xlnx-int1dx#
#=xlnx-x+C-># (don't forget the constant of integration!)