int x^r lnx dx is a 'standard' question.
For r!= -1 integrate by parts.
We don't (most of us) know the integral of lnx, but we do know its derivative, so
Let u = lnx and dv = x^r dx (in this question dv = x^(2/3) dx
With these choices, we get:
du = 1/x dx and v = int x^r dx = x^(r+1)/(r+1) (Here: 3/5x^(5/3))
int u dv = uv - int vdu = 3/5 x^(5/3) lnx - 3/5 int x^(5/3) * 1/x dx
= 3/5 x^(5/3) lnx - 3/5 int x^(2/3) dx (You'll always get the same integral fo this kind of problem.)
= 3/5 x^(5/3) lnx - 3/5 *3/5 x^(5/3)+C Or for your definite integral:
= [3/5 x^(5/3) lnx - 9/25 x^(5/3)]_1^4
Now do the arithmetic. (remember that ln1 = 0)
