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

The integral of ((ln(x))^3)/3 is 1/3 xln^3(x)-xln^2(x)+2xln(x)-2x+C

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integral of ((ln(x))^3)/3

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

\frac{1}{3} x ln^{3} (x) - x ln^{2} (x) + 2 x ln (x) - 2 x + C

Solution steps

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

Take the constant out:

\int a \cdot f (x) d x = a \cdot \int f (x) d x

= \frac{1}{3} \cdot \int (ln (x))^{3} d x

Apply Integration By Parts

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

= \frac{1}{3} (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))

= \frac{1}{3} (x ln^{3} (x) - 3 (x ln^{2} (x) - 2 (x ln (x) - x)))

Simplify

\frac{1}{3} (x ln^{3} (x) - 3 (x ln^{2} (x) - 2 (x ln (x) - x))) : \frac{1}{3} x ln^{3} (x) - x ln^{2} (x) + 2 x ln (x) - 2 x

= \frac{1}{3} x ln^{3} (x) - x ln^{2} (x) + 2 x ln (x) - 2 x

Add a constant to the solution

= \frac{1}{3} x ln^{3} (x) - x ln^{2} (x) + 2 x ln (x) - 2 x + C

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What is the integral of ((ln(x))^3)/3 ?
The integral of ((ln(x))^3)/3 is 1/3 xln^3(x)-xln^2(x)+2xln(x)-2x+C