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

The integral of (xln(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 (xln(x))^2

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

Apply exponent rule:

(a \cdot b)^{n} = a^{n} b^{n}

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

Apply Integration By Parts

= \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

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