What is the integral of x^2*log_{e}(x) ?

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

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integral of x^2*log_{e}(x)

\int x^{2} \cdot lo g_{e} (x) d x

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

Solution steps

\int x^{2} lo g_{e} (x) d x

Simplify

lo g_{e} (x) : ln (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}

Add a constant to the solution

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

x \to - 3 lim (- x^{2})

\frac{\partial}{\partial x} ((x + y + z)^{2})

\int_{0}^{4} t 16 - 3 t d t

\int 3 x^{2} 4 x^{4} d x

x \to 0 lim (4 - 5 x)

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