maths
posted by lucy on .
integrate by parts
çx (lnx)^2 dx

integrate by parts
x(lnx)^2 
∫u dv = uv  ∫v du
So in ∫x(lnx)^2 dx
let u = (lnx)^2 and dv = x dx
du/dx = 2lnx (1/x) and v = (1/2)x^2
du = (2/x)(lnx) dx and v = (1/2)x^2
∫x(lnx)^2 dx = (1/2)(x^2)(lnx)^2  ∫(1/2)x^2 (2/x)(lnx) dx
= (1/2)(x^2)(lnx)^2  ∫(x)(lnx) dx
now let's do ∫(x)(lnx) dx
let u = lnx and dv = x dx
du/dx = 1/x and v = (1/2)x^2
du = (1/x) dx and v = (1/2)x^2
∫(x)(lnx) dx = (lnx)(1/2)(x^2)  ∫(1/2)(x^2)(1/x) dx
= (1/2)(x^2)(lnx)  ∫(1/2)x dx
= (1/2)(x^2)(lnx)  (1/4)x^2
So finally ...
∫x(lnx)^2 dx
= (1/2)(x^2)(lnx)^2  ∫(x)(lnx) dx
= (1/2)(x^2)(lnx)^2  ((1/2)(x^2)(lnx)  (1/4)x^2) )
= (1/2)(x^2)(lnx)^2  (1/2)(x^2)(lnx) + (1/4)x^2)
better check my algebra, I should have written it out on paper first.