How to solve integral of y=x*ln x?

For solving the integral, we'll apply integration by
parts:

Int f(x)dx = Int x*ln
xdx

We'll apply the formula of integration by
parts:

Int udv = uv - Int
vdu

u = ln x

We'll
differentiate both sides:

du =
dx/x

dv = xdx => Int xdx = Int dv =
v

v = x^2/2

We'll substitute
u,v,du,dv in the formula above:

Int x*ln xdx = [(x^2)*(ln
x)]/2 - Int x^2dx/2x

We'll simplify and we'll
get:

Int x*ln xdx = [(x^2)*(ln x)]/2 - x^2/4 +
C

We'll factorize by
x^2/2:

Int x*ln xdx = (x^2/2)*(ln x - 1/2) +
C