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

The integral of x^2ln(x)x is 1/4 x^4ln(x)-(x^4)/(16)+C

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integral of x^2ln(x)x

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

\frac{1}{4} x^{4} ln (x) - \frac{x ^{4}}{16} + C

Solution steps

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

Simplify

x^{2} ln (x) x : x^{3} ln (x)

= \int x^{3} ln (x) d x

Apply Integration By Parts

: \frac{1}{4} x^{4} ln (x) - \int \frac{x ^{3}}{4} d x

= \frac{1}{4} x^{4} ln (x) - \int \frac{x ^{3}}{4} d x

\int \frac{x ^{3}}{4} d x = \frac{x ^{4}}{16}

= \frac{1}{4} x^{4} ln (x) - \frac{x ^{4}}{16}

Add a constant to the solution

= \frac{1}{4} x^{4} ln (x) - \frac{x ^{4}}{16} + C

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What is the integral of x^2ln(x)x ?
The integral of x^2ln(x)x is 1/4 x^4ln(x)-(x^4)/(16)+C