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)

integral

- x^{2} ln (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)

\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

in t e g r a l \frac{x ^{2}}{x ^{2} + 1}

in t e g r a l y (4 - y)^{\frac{1}{2}}

in t e g r a l (\frac{e}{2})^{x}

in t e g r a l \frac{1}{( x - a )}

in t e g r a l \frac{8 x}{( x ^{4} + 1 )}

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