How do you integrate ln(x)?

Tricky. Definitely can't do it by inspection (we don't know any fuction that just differentaties to ln(x)), it's not like we've really got anything to substitute u for if we wanted to do it by substitution and integration by parts requires two different terms being multiplied together; and we only have one! It seems like all of the ways we know how to integrate aren't going to help us much??

Maybe one of them could though. If we rewrite ln(x) = 1*ln(x) we at least have two terms in order to do integrate by parts. choosing which is u and which is dV/dx isn't going to be very hard; if we took ln(x) = dV/dx then we'd have to integrate it immediately, which was the whole problem! u = ln(x) it is then.

this gives us du/dx = 1/x and V = the integral of 1 dx = x

This looks promising. Our formula for integration by parts ( derived from the product rule) is:

Integral(udV/dx) = uV - Integral(du/dxV)       Substiting the values we just got into this gets us:

Integral(ln(x)) = xln(x) - Integral(x1/x)   Awesome. x1/x = 1 and we can definitely integrate that;

Answer = xlnx - Integral(1) = xlnx - x  = x(lnx-1) + c        [try not to forget the plus c!]

There you go. A little bit of creativity required, and we turned a seeming dead end into a complete solution! This result could definitely be useful when we're integrating more complex functions, like x/ln(x).