What is the integral of x^2(ln(x))^3 ?

The integral of x^2(ln(x))^3 is 1/3 x^3ln^3(x)-1/3 x^3ln^2(x)+2/9 x^3ln(x)-(2x^3)/(27)+C

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integral of x^2(ln(x))^3

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

\frac{1}{3} x^{3} ln^{3} (x) - \frac{1}{3} x^{3} ln^{2} (x) + \frac{2}{9} x^{3} ln (x) - \frac{2 x ^{3}}{27} + C

Solution steps

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

Simplify

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

Apply Integration By Parts

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

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

\int x^{2} ln^{2} (x) d x = \frac{1}{3} x^{3} ln^{2} (x) - \frac{2}{3} (\frac{1}{3} x^{3} ln (x) - \frac{x ^{3}}{9})

= \frac{1}{3} x^{3} ln^{3} (x) - (\frac{1}{3} x^{3} ln^{2} (x) - \frac{2}{3} (\frac{1}{3} x^{3} ln (x) - \frac{x ^{3}}{9}))

Simplify

\frac{1}{3} x^{3} ln^{3} (x) - (\frac{1}{3} x^{3} ln^{2} (x) - \frac{2}{3} (\frac{1}{3} x^{3} ln (x) - \frac{x ^{3}}{9})) : \frac{1}{3} x^{3} ln^{3} (x) - \frac{1}{3} x^{3} ln^{2} (x) + \frac{2}{9} x^{3} ln (x) - \frac{2 x ^{3}}{27}

= \frac{1}{3} x^{3} ln^{3} (x) - \frac{1}{3} x^{3} ln^{2} (x) + \frac{2}{9} x^{3} ln (x) - \frac{2 x ^{3}}{27}

Add a constant to the solution

= \frac{1}{3} x^{3} ln^{3} (x) - \frac{1}{3} x^{3} ln^{2} (x) + \frac{2}{9} x^{3} ln (x) - \frac{2 x ^{3}}{27} + C

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What is the integral of x^2(ln(x))^3 ?
The integral of x^2(ln(x))^3 is 1/3 x^3ln^3(x)-1/3 x^3ln^2(x)+2/9 x^3ln(x)-(2x^3)/(27)+C