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

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

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integral of-x^2ln(x)

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

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

Solution steps

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

Take the constant out:

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

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

Simplify

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