Integrating Log Value by Parts Method

Topic : Integration by Parts
Question : Integrate by Parts x . ln (x)

Solution :
∫ x . ln (x) dx
take u = ln (x) and dv = x dx
we know ∫ u dv = uv - ∫v du
u = ln x
dv = x dx
v = ∫dv = ∫x dx = x²/2
uv - ∫v du
= (ln x)(x²/2) - ∫x/2 . 1 dx
(u = ln x so du = 1/x dx)
= x²/2 . ln x - ∫x/2 dx + c
= x²/2 . ln x - 1/2 . x²/2
= x²/2 . ln x - x²/4 + c