Antiderivatives?

Question

What's the antiderivative of xln(x) - x? What are you supposed to do with that first x! I don't know if it's even possible to do a reverse product rule...

Answer 1

Let's break that integral in two.
The integral of -x is -x²/2.
Now we need the integral of x ln x.
Let's use parts.
U = ln x dV = x
dU = dx/x V = x²/2
So ∫ x ln x dx = x²/2 ln x -x²/4
So the final result is ½ x² ln x -3x²/4 + C =
= x²/4*(2 ln x -3) + C.