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

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

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

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

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

Solution steps

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

Take the constant out:

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

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

Add a constant to the solution

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

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