How do I integrate ln(x)?

This is a very cunning application of the integration by parts rule. Although it might look at first like integration by parts doesn't apply here since there is only the one factor, there is actually a hidden factor of 1 (since anything multiplied by 1 is itself). Thus we can set u = ln(x) and dv/dx = 1. This gives us du/dx = 1/x (since we already know how to differentiate ln(x)) and v = x. From here, we apply the integration by parts rule to get (after some rearrangement) xln(x) - x + C.Once you have seen this trick applied to ln(x), you can use it to do some other difficult integrals too.