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integrate by parts


çx (lnx)^2 dx

  • maths - ,

    integrate by parts

    x(lnx)^2

  • 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.

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