integrate with respect to x the function f(x)= xln(x)

Use integration by parts

let u=ln(x)

let dv/dx=x

therefore du/dx=1/x and v=(1/2)x^2

therefore the integral of xln(x) is equal to the following:

(1/2)x^2ln(x) - (integral with respect to x of:((1/2)x^2)/x)

= (1/2)x^2ln(x) - (integral with respect to x of:((1/2)x))

=(1/4)x^2(2ln(x)-1) + c

(I will explain further how I reached this answer during the session with provision of the whiteboard to evaluate my integrals)