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

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

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

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

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

Solution steps

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

Apply Integration By Parts

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

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

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

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

Add a constant to the solution

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

\frac{\partial}{\partial x} (e^{- (x - 3)^{2}})

\frac{d}{d y} (\frac{1}{8} y^{4} + \frac{1}{4 y ^{2}})

y = x^{2} + \frac{1}{2}, (- 7, 7)

y = - 6 x

\int (\frac{2}{x ^{3}} + x^{4}) d x

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