use integration by parts with 1 as the second function
that is make it ln(x) *1
try the following page for more explanation
mixture: integral of ln(x) using integration by parts
Don't integrate - balloontegrate!
Drawing the product rule thus...
or thus...
... (in which the straight lines differentiate, downwards, with respect to x) the challenge of integration by parts is to fill out the shape usefully but starting (unlike in differentiation) in one of the lower balloons...
As with u and v, some trial and error (or intelligent forethought, or L.I.A.T.E) may be necessary - but possibly less confusing having the picture with which to find one's way. Anyway, the right road is...
... in which the only problem is that the lower level is no longer equal to what we wanted to integrate. So correct that...
... and complete the integration ...
Balloon Calculus: worked examples from past papers