What is the integral of [ln(x)]^3 ?

The integral of [ln(x)]^3 is xln^3(x)-3(xln^2(x)-2(xln(x)-x))+C

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integral of [ln(x)]^3

\int [ln (x)]^{3} d x

x ln^{3} (x) - 3 (x ln^{2} (x) - 2 (x ln (x) - x)) + C

Solution steps

\int [ln (x)]^{3} d x

Simplify

= \int ln^{3} (x) d x

Apply Integration By Parts

: x ln^{3} (x) - \int 3 ln^{2} (x) d x

= x ln^{3} (x) - \int 3 ln^{2} (x) d x

\int 3 ln^{2} (x) d x = 3 (x ln^{2} (x) - 2 (x ln (x) - x))

= x ln^{3} (x) - 3 (x ln^{2} (x) - 2 (x ln (x) - x))

Add a constant to the solution

= x ln^{3} (x) - 3 (x ln^{2} (x) - 2 (x ln (x) - x)) + C

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What is the integral of [ln(x)]^3 ?
The integral of [ln(x)]^3 is xln^3(x)-3(xln^2(x)-2(xln(x)-x))+C