How do I get the integral of (ln(x))^2/x?

I know the answer is ln(x)^3/3 but dont know why it is.


I get that, but what happens to the x in the denominator.

2 Answers

  • Huh
    Lv 6
    6 years ago
    Favourite answer

    ∫ [((ln(x))^2 / x] dx

    Perform a u-substitution with the natural logarithm: u = ln(x)

    Thus the derivative of u: du = [1 / x] dx

    ∫ u^2 du

    Perform power rule of integration: add one to the power and divide by the new power:

    = u^3 / 3 + C

    Undo the substitution:

    = [ln(x)]^3 / 3 + C

    ... while C is the constant of integration.

  • alex
    Lv 7
    6 years ago

    u = ln(x)

    ---> integral of u^2 du = (u^3)/3 + C = (lnx)^3/3+C

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