Question

How can I evaluate the integral of xln(x)dx by using integration by parts

Answer 1

u = lnx
du = 1/x dx

dv = x dx
v = 1/2 x^2

∫u dv = uv - ∫v du, so
∫x lnx dx = 1/2 x^2 lnx - ∫(1/2 x^2)(1/x dx)
= 1/2 x^2 lnx - 1/2 ∫x dx
...