The integral of log(x) is not necessarily straight-forward. Though we can use the fact that d/dx(log(x)) = 1/x to help us.

Rather than simply trying to integrate log(x), we can use integration by parts on 1 x log(x) (as in 'one times' log(x)).

So we can differentiate the log(x) part and integrate the 1 part to give:

xlog(x) - ∫ 1 dx = xlog(x) - x

Note: if the middle step isn't clear, we can write it more explicitly as

u = log(x)  v' = 1

u' = 1/x     v = x

Where the rule for integration by parts is written as:

uv' = uv - ∫ u'v    ,  where u and v are functions of x