Its always really helpful to write out the intergral in proper notation, which will give you a dy at the end

now you can just set dv to dy, and i believe the derivative of y^ln(y) is just 0 (y^lny is a constant), correct me if i'm wrong.

so u=y^ln(y)

du=0

dv=dy

v=y

I believe by parts shouldn't be that hard after that.