Integrate, by parts, y=xln(x),

First, we need to separate the RHS as components of U and V. Using the LATE (logarithms, algebra, trigonometry, exponentials) technique, we see that logarithms have priority to algebra hence U = lnx and dV/dx = x. Next, we differentiate the U and integrate the dV/dx to obtain dU/dx = 1/x and V = x²/2. To integrate by parts we do: UV minus the integral of dU/dx times V. Thus, I (the integral) = (x²/2)lnx - ∫x/2.dx and once integrated the integral becomes (x²/2)lnx - x²/4 + C (never forget the constant C for the general solution to an integration problem).