posted by .

integrate by parts

çx (lnx)^2 dx

  • maths -

    integrate by parts


  • maths -

    ∫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.

Respond to this Question

First Name
School Subject
Your Answer

Similar Questions

  1. CALC

    i still need help on this...integrate (-2)/(xln^4(x) dx...should i be using substitution ...if yes then wut should i substitute?
  2. Calculus

    How do we integrate 1/lnx.
  3. Calculus

    How do you integrate [(x^2)(cos(2(x^3)))]?
  4. integration

    How do you integrate ; ln(lnx) and x/lnx
  5. L'Hopital's rule

    Find lim x->1+ of [(1/(x-1))-(1/lnx)]. Here is my work... =(lnx-(x-1)) / ((x-1)(lnx)) =(lnx-1) / (lnx+ (x+1)/x) This becomes(1/x) / ((1/x)+(1/x^2)) which becomes 1/ (1/x^2) This equals 1/2. I understand the answer has to be -1/2, …
  6. Calculus 1

    Find the derivative of y with respect to x. y=(x^6/6)(lnx)-(x^6/36) So far this is what I've gotten: y=(x^6/6)(lnx)-(x^6/36) y=(1/6)x^6(lnx)-(1/36)x^6 y'=(1/6)x^5(1/x)+lnx(x^5)-(1/6)x^5 What do I do now?
  7. maths

    integrate by parts integrate (4+x^2)^1/2
  8. maths

    integrate by parts integrate (4+x^2)^1/2
  9. calculus

    sorry to ask a second question so soon, but i'm just not getting this one. if f(x)= 3x lnx, then f'(x)=?
  10. Calculus

    what is the limit as h approaches 0 of ((x+h)^pi-x^pi)/h?

More Similar Questions