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

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

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integral of ln(x)*x^2

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

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

Solution steps

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

Apply Integration By Parts

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

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

\int \frac{x ^{2}}{3} d x = \frac{x ^{3}}{9}

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

Add a constant to the solution

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

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