# What is the integral of (x^2)(lnx)?

Apr 6, 2018

$\int {x}^{2} \cdot L n x \cdot \mathrm{dx} = {x}^{3} / 3 \cdot L n x - {x}^{3} / 9 + C$

#### Explanation:

After setting $\mathrm{dv} = {x}^{2} \cdot \mathrm{dx}$ and $u = L n x$ for using integration by parts, $v = {x}^{3} / 3$ and $\mathrm{du} = \frac{\mathrm{dx}}{x}$

Hence,

$\int u \mathrm{dv} = u v - \int v \mathrm{du}$

$\int {x}^{2} \cdot L n x \cdot \mathrm{dx} = {x}^{3} / 3 \cdot L n x - \int {x}^{3} / 3 \cdot \frac{\mathrm{dx}}{x}$

=${x}^{3} / 3 \cdot L n x - \int {x}^{2} / 3 \cdot \mathrm{dx}$

=${x}^{3} / 3 \cdot L n x - {x}^{3} / 9 + C$